A  TREATISE  ON 
THE  THEORY  OF  INVARIANTS 


BY 

OLIVER   E.  GLENN,  Ph.D. 

PROFESSOR    OF   MATHEMATICS    IN    THE    UNIVERSITY    OF    PENNSYLVANIA 


GINN  AND  COMPANY 

BOSTON     •     NEW   YORK     •     CHICAGO     •     LONDON 
ATLANTA     •    DALLAS     •     COLUMBUS     ■     SAN    FRANI   [SCO 


COPTRIGHT,  1915,  BY 
OLIVER  E.  GLEXN 


ALL  RIGHTS   RESERVED 
215.9 


Cftt   atbtnatum   gtegj 

GINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


PREFACE 

The  object  of  this  book  is,  first,  to  present  in  a  volume  of 
medium  size  the  fundamental  principles  and  processes  and  a 
few  of  the  multitudinous  applications  of  invariant  theory, 
with  emphasis  upon  both  the  nonsymbolical  and  the  symbol- 
ical method.  Secondly,  opportunity  has  been  taken  to  empha- 
size a  logical  development  of  this  theory  as  a  whole,  and  to 
amalgamate  methods  of  English  mathematicians  of  the  latter 
part  of  the  nineteenth  century  —  Boole,  Cay  ley,  Sylvester, 
and  their  contemporaries  —  and  methods  of  the  continental 
school,  associated  with  the  names  of  Aronhold,  Clebsch, 
Gordan,    and   Hermite. 

The  original  memoirs  on  the  subject,  comprising  an  ex- 
ceedingly large  and  classical  division  of  pure  mathematics, 
have  been  consulted  extensively.  I  have  deemed  it  expe- 
dient, however,  to  give  only  a  few  references  in  the  text.  The 
student  in  the  subject  is  fortunate  in  having  at  his  command 
two  large  and  meritorious  bibliographical  reports  which  give 
historical  references  with  much  greater  completeness  than 
would  be  possible  in  footnotes  in  a  book.  These  are  the 
article  "Invariantentheorie"  in  the  "  Enzyklopaclie  der  mathe- 
matischen  Wissenschaften"  (I  B  2),  and  W.  Fr.  Meyers 
"  Bericht  liber  den  gegenwartigen  Stand  der  Invarianten- 
theorie  "  in  the  "  Jahresbericht  der  deutschen  Mathematiker- 
Vereinigung"  for  1890-1891. 

The  first  draft  of  the  manuscript  of  the  book  was  hi  the 
form  of  notes  for  a  course  of  lectures  on  the  theory  of  inva- 
riants, which  I  have  given  for  several  years  hi  the  Graduate 
School  of  the  University  of  Pennsylvania. 

The  book  contains  several  constructive  simplifications  of 
standard  proofs  and,  in  connection  with  invariants  of  finite 


345268 


iv  THE  THEOKY  OF   [NVAKIAISTTS 

groups  of  transformations  and  the  algebraical  theory  of  ter- 
nariants,  formulations  of  fundamental  algorithms  which  may, 
it  is  hoped,  be  of  aid  to  investigators. 

While  writing   I    have  had  at   hand   and  have   frequently 
consulted  the  following  texts: 

Clebsch,  Theorie  der  binaren  Formen  (187"2). 

Clebsch,  LindemaNn,  Vorlesungen  iiber  Geometrie  (1875). 

Dickson,  Algebraic  Invariants  (lull). 

Dickson,  Madison  Colloquium  Lectures  on  Mathematics  (1913).  1.  In- 
variants and  the  Theory  of  Numbers. 

Ei.i.ioi  i.  Algebra  of  Quantics  (1895). 

F\\  i>i  Bruno,  Theorie  des  tonnes  binaires  (1876). 

Gordan,  Vorlesungen  iiber  Lnvariantentheorie  (1887). 

Gr  \<  i.  and  Young,  Algebra  of  Invariants  (  L903). 

W.  Fr.  Meyer,  Allgemeine  Formen  und  [nvariantentheorie  (1909). 

W.  Fr.  Meyer,  Apolaritat  und  rationale  Curven  (1883). 

.Salmon,  Lessons  Introductory  to  Modern  Higher  Algebra  (1859;  lilt 
ed.,  1885). 

Study,  Methoden  zur  Theorie  der  ternaren  Formen  (1889). 

().   E.  GLENN 
Phi  ladelphia,  Pa. 


CONTENTS 


CHAPTER   I.     THE   PRINCIPLES   OF   INVARIANT 
THEORY 

Section  1.     The  Nature  of  an  Invariant.     Illustrations 


I.  An  invariant  area   . 

II.  An  invariant  ratio  . 

III.  An  invariant  discriminant 

IV.  An  invariant  geometric  relation 
V.  An  invariant  polynomial 

VI.  An  invariant  of  three  lines     . 

VII.  A  differential  invariant  . 

VIII.  An  arithmetical  invariant 


PAGE 
1 

2 
4 
5 
6 
8 
9 
12 


Section  2.   Terminology  and  Definitions.    Transforma- 
tions 


I.  An  invariant 

II.  Qualities  or  forms 

III.  Linear  transformations   .... 

IV.  A  theorem  on  the  transformed  polynomial 
V.  A  group  of  transformations     . 

VI.  The  induced  group  .... 

VII.  Cogrediency     ...... 

VIII.  Theorem  on  the  roots  of  a  polynomial    . 

IX.  Fundamental  postulate  .... 

X.  Empirical  definition         .... 

XL  Analytical  definition       .... 

XII.  Annihilators    ...... 


14 
14 
15 
16 
18 
19 
20 
21 
22 
22 
23 
25 


Section  3.     Special  Invariant  Formations 

I.     Jacobians         ..... 
II.     Hessians ..... 
III.     Binary  resultants    .... 


27 
28 
29 


vi  THE  THEORY  OE  INVARIANTS 

PAGE 

IV.     Discriminant  of  a  binary  form  31 

V.     Universal  covariants 32 


CHAPTER  II.     PROPERTIES   OF   INVARIANTS 

Section  1.     Homogeneity  of  a  Binary  Concomitant 

I.     Theorem  on  homogeneity 33 

Section  2.     Index,  Order,  Degree,  Weight 

I.     Definitions 35 

II.     Theorem  on  the  index 35 

III.     Theorem  on  weight 36 

Section  3.     Simultaneous  Concomitants 

I.     Theorem  on  index  and  weight 38 

Section  4.  Symmetry.  Fundamental  Existence  Theorem      39 

CHAPTER  III.     THE   PROCESSES   OF   INVARIANT    THEORY 

Section  1.     Inyariant  Operators 

I.  Polars 42 

II.  Polar  of  a  product 45 

III.  Aronhold's  polars 46 

IV.  Modular  polars  . 48 

V.  Operators  derived  from  the  fundamental  postulate         .         .  49 

VI.  Trans  vection 51 

Section  2.       The   Aronhold    Symbolism.      Symbolical 
Processes 


I.  Symbolical  representation 

II.  Symbolical  polars 

III.  Symbolical  transvectants 

IV.  Standard  method  of  transvection 
V.  Formula  for  the  rth  transvectant 

VI.  Special  cases  of  operation  by  12 

VII.  Fundamental  theorem  of  symbolical  theory 


53 
55 
56 

57 
59 
61 
62 


CONTENTS  vii 

Section    3.      Deducibility.     Elementary    Irreducible 
Systems 

PAliE 

I.     Illustrations 64 

II.     Reduction  by  identities 66 

III.     Concomitants  of  binary  cubic.     Table  I 68 

Section  4.     Concomitants  in  Terms  of  the  Roots 

I.  Theorem  on  linear  factors 69 

II.     Conversion  operators 70 

III.  Principal  theorem 72 

IV.  Hermite's  reciprocity  theorem 76 

Section  5.    Geometrical  Interpretations.    Involution 

I.     Involution 78 

II.  Projective  properties  represented  by  vanishing  covariants      .  80 

CHAPTER   IV.     REDUCTION 

Section  1.     Gordan's  Series.     The  Quartic 

I.     Gordan's  series 83 

II.     The  quartic.     Table  II .  89 

Section  2.     Theorems  on  Transvectants 

I.     Monomial  concomitant  a  term  of  a  transvectant      ...  92 
II.     Theorem  on  the  difference  between  two  terms  of  a  trans- 
vectant        .                                            94 

III.     Difference  between  a  transvectant  and  one  of  its  terms  .         .  96 

Section  3.     Reduction  of  Transvectant  Systems 

I.     Reducible  transvectants  of  a  special  type          ....  97 

II.     Fundamental  systems  of  cubic  and  quartic  forms    .         .         .  100 

III.     Reducible  transvectants  in  general 102 


Section  4.     Syzygies 

I.  Reducibility  of  ((/,  g),  h) 

II.  Product  of  two  Jacobians  . 

III.  The  square  of  a  Jacobian  . 

IV.  Syzygies  for  the  cubic  and  quartic  forms 
V.  Syzygies  derived  from  canonical  forms 


105 

106 
107 
107 
108 


viii  THE  THEORY  OE  INVARIANTS 

Skitkix  .").     Humbert's  Theorem 

PAGE 

I.     Theorem .        .112 

II.     Linear  diophantine  equations    .......  lltj 

III.     Finitene8S  of  system  of  syzygies 119 

Section  6.     Jordan's   Lemma 

I.     Representation  of  a  form 120 

II.     Jordan's  lemma  .........  122 

Section  7.     Grade 

I.     Definition  ...........  124 

II.     Grade  of  a  covariant 124 

III.  Covariant  congruent  to  one  of  its  terras   .....  1l'.~> 

IV.  Representation  of  a  covariant  of  a  covariant    ....  126 

CHAPTER   V.     GORDAX'S   THEOREM 

Section  1.     Proof  of  the  Theorem 

I.     Lemma  1 128 

II.     Lemma  2 131 

III.  Lemma  3 133 

IV.  Theorem 138 

Section  2.     Fundamental  Systems  of  Cubic  and  Quartic 

I.     System  of  cubic 141 

II.     System  of  quartic 142 

CHAPTER   VI.     FUNDAMENTAL   SYSTEMS 

Se(  tion   1.     Simultaneous  Systems 

I.     Linear  form  and  quadratic 144 

II.     Linear  form  and  cubic .         .145 

III.  Two  quadratics 145 

IV.  Quadratic  and  cubic.     Table  III 146 

Section  2.     System  of  the  Quintic 

I.     The  quintic.     Table  IV -  150 


CONTENTS  ix 

Section  3.     Resultants  in  Aronhold's  Symbols 

PAGE 

I.     Linear  form  and  n-ic 151 

II.     Quadratic  form  and  n-ic 151 

Section  4.     Fundamental  Systems  for  Special  Groups 

I.     Boolean  system  of  a  linear  form 156 

II.     Boolean  system  of  a  quadratic  ......     156 

III.     Formal  modular  system  of  a  linear  form  .         .         .         .157 

Section  5.     Associated  Forms 158 


CHAPTER  VII.   COMBINANTS  AND  RATIONAL  CURVES 
Section  1.     Combinants 


I.  Definition  .... 

II.  Theorem  on  Aronhold  operators 

III.  Partial  degrees 

IV.  Resultants  are  combinants 

V.  Bezout's  form  of  the  resultant 


162 
163 
165 

166 
168 


Section  2.     Rational  Curves 

I.     Meyer's  translation  principle 
II.     Covariant  curves 


169 
171 


CHAPTER  VIII. 


SEMIN  VARIANTS.     MODULAR 
INVARIANTS 


Section  1.     Binary  Seminvariants 

I.  Generators  of  binary  group 

II.  Definition .... 

III.  Theorem  on  annihilator  D 

IV.  Formation  of  seminvariants 

V.  Roberts'  theorem 

VI.  Symbolical  representation  of  seminvariants 

VII.  Finite  systems  of  seminvariants 


175 

176 
176 
177 
179 
180 
184 


Section  2.     Ternary  Seminvariants 

I.     Annihilators     ...... 

II.     Symmetric  functions  of  groups  of  letters 


189 
191 


x  THE  THEORY  OF  INVARIANTS 

PAOB 

III.  Semi-discriminants.     Table  V 193 

IV.  Invariants  of  m-lines 202 

Section  3.     Modular  Invariants  and  Covariants 

I.     Fundamental  system  of  modular  quadratic  form,  modulo  3. 

Table  VI 203 

CHAPTER  IX.  INVARIANTS  OF  TERNARY  FORMS 

Section  1.     Symbolical  Theory 

I.  Polars  and  transvectants 209 

II.  Contragrediency         .         .       • 212 

III.  Fundamental  theorem  of  symbolical  theory      ....  213 

IV.  Reduction  identities 218 

Section  2.     Transvectant  Systems 

I.     Theorem  on  monomial  concomitants 219 

II.     The  difference  between  two  terms  of  a  transvectant        .         .     220 

III.  Fundamental   systems   of    invariant    formations    of    ternary 

quadratic  and  cubic  forms.     Table  VII     ....     223 

IV.  Fundamental  system  of  two  ternary  quadrics  ....     225 

Section  3.     Clebsch's  Translation  Principle  .        .        .    228 

APPENDIX 

Exercises  and  theorems 231-211 

INDEX 243 


THE  THEORY   OF  INVARIANTS 

CHAPTER   I 

THE    PRINCIPLES   OF   INVARIANT   THEORY 

SECTION    1.     THE    NATURE    OF    AN    INVARIANT. 
ILLUSTRATIONS 

We  consider  a  definite  entity  or  system  of  elements,  as  the 
totality  of  points  in  a  plane,  and  suppose  that  the  system  is 
subjected  to  a  definite  kind  of  a  transformation,  like  the 
transformation  of  the  points  in  a  plane  by  a  linear  trans- 
formation of  their  coordinates.  Invariant  theory  treats  of 
the  properties  of  the  system  which  persist,  or  its  elements 
which  remain  unaltered,  during-  the  changes  which  are  im- 
posed upon  the  system  by  the  transformation. 

By  means  of  particular  illustrations  we  can  bring  into 
clear  relief  several  defining  properties  of  an  invariant. 

I.  An  invariant  area.  Given  a  triangle  ABO  drawn  in 
the  Cartesian  plane  with  a  vertex  at  the  origin.  Suppose 
that  the  coordinates  of  A  are  (xv  y-^);  those  of  B  (xv  y^). 
Then  the  area  A  is 

A  =  JOi^a  -  *2#l)' 
or,  in  a  convenient  notation, 

A  =  \(xy). 

Let  us  transform  the  system,  consisting  of  all  points  in  the 
plane,  by  the  substitutions 

x  =  x^'  +  ^y\  y  =  V'  +  fHSf'- 


•2  THE    THEORY    OF    INVARIANTS 


The  area  of  the  triangle  into  which  A  is  then  carried  will  be 

A'=K^-4#i)  =  K*y)5 

and  by  applying  the  transformations  directly  to  A, 

A=(V2"Vl)A'  C1) 

If  we  assume  that  the  determinant  of  the  transformation  is 

lll,i,-v'  2>  =  (X/0=1, 

then  A'  =  A. 

Z%ws  (Ag  area  A  of  the  triangle  ABC  remains  unchanged 
under  a  transformation  of  determinant  unity  and  is  an  in- 
variant of  the  transforma- 
tion. The  triangle  itself  is 
not  an  invariant,  but  is  car- 
ried into  abC.  Tiie  area 
A  is  called  an  absolute  in- 
variant if  D  =  1 .  If  I)  =t  1, 
all  triangles  having  a  vertex 
at  the  origin  will  have  their 
areas  multiplied  by  the  same 
number  D'1  under  the  trans- 
formation. In  such  a  case 
A  is  said  to  be  a  relative  invariant.  The  adjoining  figure 
illustrates  the  transformation  of  ^4.(5,  (5),  -6(4,  6),  (7(0,  0)  by 
means  of  x  =  ^  +  y\  y  =  x' +  2y' . 

II.  An  invariant  ratio.  In  I  the  points  (elements)  of  the 
transformed  system  are  located  by  means  of  two  lines  of 
reference,  and  consist  of  the  totality  of  points  in  a  plane.  For 
;i  second  illustration  we  consider  the  system  of  all  points  on 
a  line  EF. 

We  Locate  a  point  C  on  this  line  by  referring  it  to  two 
fixed  points  of  reference  P,  Q.  Thus  C  will  divide  the 
segment  PQ  in  a  definite  ratio.     This  ratio, 

PO/CQ, 


THE   PRINCIPLES    OF   INVARIANT   THEORY          3 

is  unique,  being  positive  for  points  C  of  internal  division  and 
negative  for  points  of  external   division.     The  point   0  is 

*. -£ 2 2 £ ,F 

said  to  have  for  coordinates  any  pair  of  numbers  (xv  x2~) 
such  that  „.        r>n 

x2       CQ 

where  X  is  a  multiplier  which  is  constant  for  a  given  pair  of 
reference  points  P,  Q.  Let  the  segment  PQ  be  positive  and 
equal  to  /a.  Suppose  that  the  point  C  is  represented  by  the 
particular  pair  (pv  p2~),  and  let  D(qv  q2)  be  any  other  point. 
Then  we  can  find  a  formula  for  the  length  of  CD.     For, 

CQ  =  PO  =      PQ  fi 

Pi  XPl        XPl+2>2        XPl+p2 

and  D  Q  _       /x 

%       M\  +  % 
Consequently 

CD=CQ-DQ  = ^^ (3) 

(Xq1  +  q2)(^p1+p2) 

Theorem.      The  anharmonic  ratio  \  CDEF\  of  four  points 
C(PvPz)'  Di(lv  &)'  E(rv  r2>'  F(sv  «2>'  defined  by 

\CDEF\=CD'EF, 
^      CF.ED 

is  an  invariant  under  the  general  linear  transformation 

T :  xx=  Xjajj  +  fxxx'2,  x2  =  X2x[  +  f*2z2,  (X/x)  =£  0.  (30 

In  proof  we  have  from  (3) 

\CDEF\  =  (W)(sr^. 
(sp)  (qr) 

But  under  the  transformation  (cf.  (1)), 

(qp~)  =  (\n)(q'p'\  (4) 


4  THE   THEORY   OF   INVARIANTS 

and  so  on.     Also,  C,  D,  E,  F  are  transformed  into  the  points 

respectively.     Hence 

(V)(«r)      (s'p'Xq'r')  J' 

and  therefore  the  anharmonic  ratio  is  an  absolute  invariant. 

III.  An  invariant  discriminant.  A  homogeneous  quadratic 
polynomial, 

f  =  a0x21  +  2  a^c^c%  +  a2^, 

when  equated  to  zero,  is  an  equation  having  two  roots  which 
are  values  of  the  ratio  x1/x2.  According  to  II  we  may  repre- 
sent these  two  ratios  by  two  points  C(pv  jt?2),  J)(qv  q%)  on 
the   line  EF.     Thus  we    may  speak  of    the    roots  (/>r  p2~), 

(qv  ?2)  of/- 

These  two  points  coincide  if  the  discriminant  of/  vanishes, 
and  conversely  ;  that  is  if 

J)  =  4(#0a2  —  a\ )  =  0. 

If  /  be  transformed  by  T,  the  result  is  a  quadratic  poly- 
nomial in  x'v  x'v  or 

Now  if  the  points  .C,  D  coincide,  then  the  two  transformed 
points  C,  D'  also  coincide.  For  if  CD=  0,  (3)  gives  (qp) 
=  0.  Then  (4)  gives  (q'p'}  =  0,  since  by  hypothesis  (Xfi)  =^0. 
Hence,  as  stated,  CD'  =  0. 

It  follows  that1  the  discriminant  D'  of  f  must  vanish  as  a 
consequence  of  the  vanishing  of  D.      Hence 

D'  =  KB. 

The  constant  iTmay  be  determined  by  selecting  in  place 
of  /  the  particular  quadratic  fx  =  2  xtx2  for  which  D  =  —  4. 
Transforming  fx  by  T  we  have 

/J  =  -1  X:  Vi  +  2(X1/Lt2  +  X^zfr  +  2  fi^x*  ; 


THE   PRINCIPLES   OF  INVARIANT   THEORY         5 

and  the  discriminant  of  f[  is  D'  =  —  4(X/u)2.  Then  the  sub- 
stitution of  these  particular  discriminants  gives 

-4(V)2  =  -4JT, 

We  may  also  determine  -ST  by  applying  the  transformation  T 
to/ and  computing  the  explicit  form  of/'.     We  obtain 

a'0  =  «0Xf  +  2  ajXjA-2  +  «2X|, 

a[  =  (IqX^  +  ^(X^  +  X^)  +  a2X^v  (5) 

and  hence  by  actual  computation, 

4(a'0a'2  -  at)  =  4(XA02(a0«2  -  a?), 

or,  as  above, 

D>  =  (X/i)2i>. 

Therefore  the  discriminant  of/ is  a  relative  invariant  of  T 
(Lagrange  1773)  ;  and,  in  fact,  the  discriminant  of  f  is 
always  equal  to  the  discriminant  of  /  multiplied  by  the 
square  of  the  determinant  of  the  transformation. 

Preliminary  Geometrical  Definition.  If  there  is 
associated  with  a  geometric  figure,  a  quantity  which  is  left 
unchanged  by  a  set  of  transformations  of  the  figure,  then  this 
quantity  is  called  an  absolute  invariant  of  the  set  (Halphen). 
In  I  the  set  of  transformations  consists  of  all  linear  trans- 
formations for  which  (X/a)  =  1.  In  II  and  III  the  set  consists 
of  all  for  which  (X/i)  =£  0. 

IV.  An  invariant  geometrical  relation.     Let  the  roots  of 
the    quadratic   polynomial  /  be  represented  by  the  points 
CPvPi)'  0*i'  r^)->  and  let  $  De  a  second  polynomial, 
(f)  =  bQx\  +  2  b^cxx2  +  b2x\\ 

whose  roots  are  represented  by  (qv  q2~),  (sv  s2),  or,  in  a 
briefer  notation,  by  (5-),  (s).  Assume  that  the  anharmonic 
ratio  of  the  four  points  (/?),  (<?),  (>),  (s),  equals  minus  one, 


6  THE   THEORY   OF    INVARIANTS 

(qp)(sr)  =  _1 
(sp)(qr) 

The  point  pairs  f=  0,  $  =  0  are  then  said  to  be  harmonic 
conjugates.      We  have  from  (6) 

2  h  =  2  2VVi?i  +  '2PirihQ2  ~  Oir2  t  P2ri)('hs2  +  fhsi)  =  °- 

/=  O1P2  -  ^DOVa  -  a'2ri)' 

Hence 


a0  = 


p2r2,  2a1  =  -(  p2r1+p)r2),  a2=p1rv 


h  =  ?2S2'   2  &1  =  —  (  ?2S1  +  'llS2  >'      &2  =  ?1«1» 

and  by  substitution  in  (2  A)  we  obtain 

A  =  a0b2  -  2  fljftj  +  a2b0  =  0.  (7) 

That  h  is  a  relative  invariant  under  T  is  evident  from  (6): 
for  under  the  transformation/,  <£  become,  respectively, 

/'  =  (Ap'i  -  Ap'i  K*'A  -  X'-A^ 

Pi  =  Vi  -  /*i^2'  P-i  =  -\lh  +  \Pv 

r'l  =  /Vl  —  /VV    r2  =  —  X2ri  +  Xlr2' 

Hence 

Thatis'  h'  =  (\fiyh. 

Therefore  the  bilinear  function  It  of  the  coefficients  of  two 
quadratic  polynomial*,  representing  the  condition  that  their 
root  pairs  be  harmonic  conjugates,  is  a  relative  invariant  of  the 
transformation  T.  It  is  sometimes  called  a  joint  invariant, 
or  simultaneous  invariant  of  the  two  polynomials  under  the 
transformation. 

V.  An  invariant  polynomial.  To  the  pair  of  polynomials 
/.  4>.  let  a  third  quadratic  polynomial  be  adjoined, 

y}r  =  V'i  +  2  6V'i-r2  +  'Vt> 

=  (xxU2  —  »2%l)(a:lv2  —  X2V1  '• 


THE   PRINCIPLES   OF  INVARIANT   THEORY 


Let  the  points  (uv  w2)  (_vv  v2),  be  harmonic  conjugate  to  the 
pair  (j9),  (r);  and  also  to  the  pair  (<?),  (s).     Then 

coh  -  2  cA  +  6A    =  °> 

0     1  "t-        ^112  "i-      2    2  ^~ 

Elimination  of  the  c  coefficients  gives 


(7  = 


»i 


•C'o  X1X1 


1*2 


0. 


(8) 


This  polynomial, 

0=  (a0b1  -  axb^x\  +  (a0b2  -  «260)a:1a-2  +  (ax62  -  aj>{)x% 

is  the  one  existent  quadratic  polynomial  whose  roots  form 
a  common  harmonic  conjugate  pair,  to  each  of  the  pairs/,  <f>. 
We  can  prove  readily  that  0  is  an  invariant  of  the  trans- 
formation T.     For  we  have  in  addition  to  the  equations  (5), 

b'0  =  b0\j  +  2  ^W  +  52Xf, 

b[  =  fyjXj/ij  +  ^(Xj/Ug  +  X^)  +  i2X2/z2,  (9) 

Also  if  we  solve  the  transformation  equations  T  for  a/j,  x'2  in 
terms  of  2^,  xv  we  obtain 

«i  =  (X/*)_1(/Vi  -  A*i«a)>  (10) 

4  =  (x/*)_1(—  XgiCj  +  x^2). 

Hence  when  /,  <£  are  transformed  by  T,  (7  becomes 

C'  = 

\  («0Xf  -f  2  a^X^  +  «2X|)  [^Xj/ij  +  ^(X^ +  X2^ )  +  62X2/z2] 
-  (b0\\  +  2  JjXjXa  +  62Xf )  OqX^j  +  aiCXj/ig  +  X^)  +  a2X2^2]  } 
X(X/x)-2(^1_/ila-2)2+-.  (11) 

When    this  expression  is  multiplied  out  and  rearranged  as 
a  polynomial  in  xv  x2,  it  is  found  to  be  (X/i)  C.     That  is, 

and  therefore  C  is  an  invariant. 


THE   THEORY   OF   INVARIANTS 


It  is  customary  to  employ  the  term  invariant  to  signify 
a  function  pf  the  coefficients  of  a  polynomial,  which  is  left 
unchanged,  save  possibly  for  a  numerical  multiple,  when  the 
polynomial  is  transformed  by  T.  If  the  invariant  function 
involves  the  variables  also,  it  is  ordinarily  called  a  covariant. 
Thus  D  in  III  is  a  relative  invariant,  whereas  C  is  a  relative 
covariant. 

The  Inverse  of  a  Linear  Transformation.  The 
process  (11)  of  proving  by  direct  computation  the  invari- 
ancy  of  a  function  we  shall  call  verifying  the  invariant  or 
covariant.  The  set  of  transformations  (10)  used  in  such  a 
verification  is  called  the  inverse  of  T  and  is  denoted  by  T~l. 

VI.  An  invariant  of  three  lines.  Instead  of  the  Cartesian 
coordinates  employed  in  I  we  may  introduce  homogeneous 
variables  (xv  xv  rr3)  to  represent  a  point  P  in  a  plane. 
These  variables  may  be  regarded  as  the  respective  distances 
of  P  from  the  three  sides  of  a  triangle  of  reference. 
Then  the  equations  of  three  lines  in  the  plane  may  be  written 

^11^1    '    ^12*^2    '    ^13X3  =  "» 
^21"C1    '    ^22"^2  ~>"  ^23*^3  =      ' 


aZlXl  +  aMp9  +  a 


The  eliminant  of  these, 


D  = 


'ii 


'21 


'31 


"12 


22 


82 


33l(3 


13 


=  0. 


l23 


'33 


evidently  represents  the  condition  that  the  lines  be  concur- 
rent. For  the  lines  are  concurrent  if  D  =  0.  Hence  we 
infer  from  the  geometry  that  D  is  an  invariant,  inasmuch  as 
the  transformed  lines  of  three  concurrent  lines  by  the  fol- 
lowing transformations,  S,  are  concurrent : 

pj  +  [l^  +  Vxx'^ 

S:     x0  =  Xo^i  +  ^2^2  +  v2x'v    ( ^-A^)  H-  "•  (12) 

•;  +  fi3.v'2  +  v3x'r 


xl 

= 

A-jZ 

•''■J 

AiyX 

•r3 

= 

A3.r 

THE   PRINCIPLES   OF   INVARIANT   THEORY 


To  verify  algebraically  that  D  is  an  invariant  we  note  that 
the  transformed  of 

ai\x\  +  ai1x1  +  ai3X3   (*'  =  !»  2'  3)> 
by  #  is 

(aaX1  +  ai2\2  +  tf/3*.3Vi  +  Oa/h  +  a/2^2  +  'Ws)2?  +  (%yi 
+  W  +  Vs>3  (*  =  X>  2'  3)-  (13> 

Thus  the  transformed  of  D  is 

an\  +  tf12\2  +  au\     an/j,1  +  rr12/x2  +  alsp3 

D'  =    a21X.1  +  a22X2  +  a23X3       «21/*!  +  «22^2  +  a23^3 
•         «31X1  +  «32X2  +  «33X3       ^31^1  +  «32^2  +  a33^3 

anvx  +  a12v2  +  auvz 

ai\Vl  +  a22V1  +  a23y3 

Vl  +  HlV1  +  a33y3 
=  (\/J,v)D.  (14) 

The  latter  equality  holds  by  virtue  of  the  ordinary  law  of  the 
product  of  two  determinants  of  the  third  order.  Hence  D  is 
an  invariant. 

VII.  A  differential  invariant.  In  previous  illustrations 
the  transformations  introduced  have  been  of  the  linear 
homogeneous  type.  Let  us  next  consider  a  type  of  trans- 
formation which  is  not  linear,  and  an  invariant  which  repre- 
sents the  differential  of  the  arc  of  a  plane  curve  or  simply 
the  distance  between  two  consecutive  points  (x,  «/)  and 
(x  +  dx,  y  +  dy}  in  the  (x,  y)  plane. 

We  assume  the  transformation  to  be  given  by 

x'  =  X(x,  y,  a),  y'  =  Y(x,  y,  a), 

where  the  functions  X,  Y  are  two  independent  continuous 
functions  of  x,  y  and  the  parameter  a.  We  assume  (a)  that 
the  partial  derivatives  of  these  functions  exist,  and  (J)  that 


10  THE   THEORY  OF   INVARIANTS 

these  are  continuous.  Also  (c)  we  define  X,  Y  to  be  such 
that  when  a  =  a0 

X(x,  y,  a0)  =  x,    Y(x,  y,  a0)  =  y. 

Then  let  an  increment  8a  be  added  to  a0  and  expand  each 
function  as  a  power  series  in  8a  by  Taylor's  theorem.  This 
gives 

»y,     '      ^  (15) 

y'  =  Y(x,  y,  a0)  4-  - — ^— ^ — ¥  8a  +  .... 

Since  it  may  happen  that  some  of  the  partial  derivatives  of 
X,  Y  may  vanish  for  a  =  a0,  assume  that  the  lowest  power 
of  8a  in  (15)  which  has  a  non-vanishing  coefficient  is  (Sa)*, 
and  write  (8a)*  =  8t.  Then  the  transformation,  which  is  in- 
finitesimal, becomes 

j     x  =  x  +  £8t, 

y'  =  y  +  v8t. 

where  £,  t]  are  continuous  functions  of  x,  y.  The  effect  of 
operating /upon  the  coordinates  of  a  point  P  is  to  add  infin- 
itesimal increments  to  those  coordinates,  viz. 

8x  =  |&, 

8y  =  r)8t. 

Repeated  operations  with  I  produce  a  continuous  motion 
of  the  point  P  along  a  definite  path  in  the  plane.  Such  a 
motion  may  be  called  a  stationary  streaming  in  the  plane 
(Lie). 

Let  us'  now  determine  the  functions  £,  77,  so  that 

a  =  da?  +  dy2 

shall  be  an  invariant  under  7. 

By  means  of  /,  a  receives  an  infinitesimal  increment   8cr. 
In  order  that  a  may  be  an  absolute  invariant,  we  must  have 

\  8a-  =  dxhdx  4-  dyhdy  =  0, 


THE   PRINCIPLES   OF   INVARIANT   THEORY      11 

or,  since  differential  and  variation  symbols  are  permutable, 

dxdhx  +  dydhy  =  dxd%  +  dydrj  =  0. 
Hence 

(£xdx  +  %ydy~)dx  +  Qqxdx  +  r)vdy)dy  =  0. 

Thus  since  dx  and  dy  are  independent  differentials 
That  is,  £  is  free  from  a:  and  rj  from  ?/.     Moreover 

bij  Vxx  £j/;/  "• 

Hence  £  is  linear  in  i/,  and  tj  is  linear  in  x ;  and  also  from 

f„  =  -  ??*< 
f  =  ay  +  &   ?;  =  —  az  +  7.  (17) 

Thus  the  most  general  infinitesimal  transformation  leaving 
cr  invariant  is 

I:x'  =x  +  (ay  +  /3)  St,  yl  =  y  +  (-  arc  +  7)^.  (18) 

Now  there  is  one  point  in  the  plane  which  is  left  invari- 
ant, viz. 

x  =  7/«,  y  =  —  fi/a. 

The  only  exception  to  this  is  when  a  =  0.  But  the  trans- 
formation is  then  completely  defined  by 

x'  =  x  +  /38t,  y'  =  y  +  ySt, 

and  is  an  infinitesimal  translation  parallel  to  the  coordinate 
axes.  Assuming  then  that  a =£  0,  we  transform  coordinate 
axes  so  that  the  origin  is  moved  to  the  invariant  point. 
This  transformation, 

x  =x  +  7/0,  y  =  y-  £/«, 

leaves  a  unaltered,  and  /becomes 

x'  =x  +  «?/&,  y'  =  y  —  ax8t.  (19) 

But  (19)  is  simply  an  infinitesimal  rotation  around  the 
origin.  We  may  add  that  the  case  a  =  0  does  not  require  to 
be  treated  as  an  exception  since  an  infinitesimal  translation 


12 


THE   THEORY   OF   INVARIANTS 


may  be  regarded  as  a  rotation  around  the  point  at  infinity. 
Thus, 

Theorem.  The  most  general  infinitesimal  transformation 
which  leaves  a  =  dx2  +  dy2  invariant  is  an  infinitesimal  rota- 
tion around  a  definite  invariant  point  in  the  plane. 

We  may  readily  interpret  this  theorem  geometrically  by 
noting  that  if  a  is  invariant  the  motion  is  that  of  a  rigid 
figure.  As  is  well  known,  any  infinitesimal  motion  of  a  plane 
rigid  figure  in  a  plane  is  equivalent  to  a  rotation  around  a 
unique  point  in  the  plane,  called  the  instantaneous  center. 
The  invariant  point  of  I  is  therefore  the  instantaneous  center 

of   the   infinitesi- 
mal rotation. 

The  adjoining 
figure  shows  the 
invariant  point 
( (?)  when  the 
moving  figure  is 
a  rigid  rod  R  one 
end  of  which  slides  on  a  circle  S,  and  the  other  along  a 
straight  line  L.  This  point  is  the  intersection  of  the  radius 
produced  through  one  end  of  the  rod  with  the  perpendicular 
to  L  at  the  other  end. 

VIII.  An  arithmetical  invariant.  Finally  let  us  intro- 
duce a  transformation  of  the  linear  type  like 

T  :  xx  =  X^  +  /*!4*  x2  =  \x'i  +  /V?' 
but  one  in  which  the  coefficients  X,  fi  are  positive  integral 
residues  of  a  prime  number  p.     Call  this  transformation  Tp. 
We  note  first  that  Tp  may  be  generated  by  combining  the 
following  three  particular  transformations  : 

(5)   x1  =  x'v  x%  —  \x'v  (20) 

(  Cj     X^  =  X<p  .T2  =         2^1 


THE   PRINCIPLES    OF    INVARIANT   THEORY       13 

where  U  A.  are  any  integers  reduced  modulo  p.     For    (a) 
repeated  gives 

Repeated  r  times  (a)  gives,  when  rt  =  u  (mod  jt>), 

Then  (<?)  combined  with  (d)  becomes 

Proceeding  in  this  way  Tp  may  be  built  up. 


where  the  coefficients  are  arbitrary  variables ;   and 

g  =  a0x\  +  a^xp^  +  a;^)  +  ^4^  v  (21> 

and  assume  p  =  3.  Then  we  can  prove  that  g  is  an  arith- 
metical co variant ;  in  other  words  a  co variant  modulo  3. 
This  is  accomplished  by  showing  that  if  /  be  transformed 
by  Tz  then  g'  will  be  identically  congruent  to  g  modulo  3. 
When  f  is  transformed  by  (•<?)  we  have 

That  is, 

a'()  =  a2,  a  J  =  —  av  a'2  =  a0. 

The  inverse  of  (e)  is  x2  =  xv  x'1  =  —  x2.     Hence 

g'  =  a2x\  4-  a^x^i  4-  ^'2)  +  a0x\  =g, 

and  g  is  invariant,  under  (<?)•/ 

Next  we  may  transform /by  (a)  ;  and  we  obtain 

a'Q  =  a0,  a\  —  a0t  +  «x,  #2  =  rt</2  +  ^  ait  +  a2- 
The  inverse  of  (a)  is 

wIsa    *'   .>^      •/-'I     •t'-l  i/tL-'fy% 

Therefore  we  must  have 

g'  =  a0(x1  -  tx2y  +  (a0<  +  ax)  [(^  -  te2)3*2  +  («i  -  te2)4] 

+  (a0*2  4-  2  a/  +  a2)4  (22) 

=  «0a^  4-  ax{x\x2  +  x^x^)  +  a2x\  (mod  3). 


14  THE    THEORY   OF   INVARIANTS 

But  this  congruence  follows   immediately  from  the  follow- 
ing case  of  Fermat's  theorem  : 

t?  =  t  (mod  3). 

Likewise  g  is  invariant  with  reference  to  (6).     Hence  g  is 
a  formal  modular  covariant  of/  under  Tz. 

SECTION  2.     TERMINOLOGY     AND     DEFINITIONS.     TRANS- 
FORMATIONS 

We  proceed  to  formulate  some  definitions  upon  which 
immediate  developments  depend. 

I.  An  invariant.  Suppose  that  a  function  of  n  variables, 
/,  is  subjected  to  a  definite  set  of  transformations  upon 
those  variables.  Let  there  be  associated  with  /  some  defi- 
nite quantity  <£  such  that  when  the  corresponding  quantity 
</>'  is  constructed  for  the  transformed  function/'  the  equality 

<f>'=M<f> 

holds.  Suppose  that  M  depends  only  upon  the  transforma- 
tions, that  is,  is  free  from  any  relationship  with/.  Then  $ 
is  called  an  invariant  of  /under  the  transformations  of  the  set. 
The  most  extensive  subdivision  of  the  theory  of  invariants 
in  its  present  state  of  development  is  the  theory  of  invari- 
ants of  algebraical  polynomials  under  linear  transformations. 
Other  important  fields  are  differential  invariants  and  num- 
ber-theoretic invariant  theories.  In  this  book  we  treat,  for 
the  most  part,  the  algebraical  invariants. 

II.  Quantics  or  forms.  A  homogeneous  polynomial  in  n 
variables  xv  .r2.  •••,  xn,  of  order  m  in  those  variables  is  called  a 
quantic,  or  form,  of  order  m.      Illustrations  are 

f(xv  .r2)  =  a0rf  +  3  avrp-2  +  3  a^c^\  +  azx% 
J\xv  xv  xz)  =  a20(rr!  "+"  -  aivyc\xi  ~t"  aoiox2  "I"  '"  ('iorrrr3 

+  -  ('oii-r2-r3  ~^~  aW&XZ' 

With  reference  to  the  number  of  variables  in  a  quantic  it 


THE    PRINCIPLES   OF   INVARIANT   THEORY       15 

is  called  binary,  ternary  ;  and  if  there  are  n  variables, 
w-ary.  Thus  f(xv  x2)  is  a  binary  cubic  form ;  f(xv  x2,  x3)  a 
ternary  quadratic  form.  In  algebraic  invariant  theories  of 
binary  forms  it  is  usually  most  convenient  to  introduce  with 

each  coefficient  a{  the  binomial  multiplier  (  .  J,  as  in  f(_zv  x2). 

When  these  multipliers  are  present,  a  common  notation  for  a 
binary  form  of  order  m  is  (Cayley) 

/Or  »2)=  Oh*  av  •"■>  amlzv  a-2)m  =  «0zf  +  maxz^-xz2  -\ . 

If  the  coefficients  are  written  without  the  binomial  numbers, 
we  abbreviate 

f(xv  x2~)  =  (a0,  av  ••-,  am\xv  x2~)m  =  a^zf  +  axz^z%  -\ . 

The  most  common  notation  for  a  ternary  form  of  order  m  is 
the  generalized  form  of  f(xv  xv  a*3)  above.     This  is 

V       lm 

p,<i,r=o  [P\q\r 

where  p,  q,  r  take  all  positive  integral  values  for  which 
p  +  q  +  r  =  m.  It  will  be  observed  that  the  multipliers 
associated  with  the  coefficients  are  in  this  case  multinomial 
numbers.  Unless  the  contrary  is  stated,  we  shall  in  all  cases 
consider  the  coefficients  a  of  a  form  to  be  arbitrary  variables. 
As  to  coordinate  representations  we  may  assume  (zv  z2,  xs~), 
in  a  ternary  form  for  instance,  to  be  homogenous  coordi- 
nates of  a  point  in  a  plane,  and  its  coefficients  apqr  to  be 
homogenous  coordinates  of  planes  in  Jf-space,  where  M+  1 
is  the  number  of  the  «'s.  Thus  the  ternary  form  is  repre- 
sented by  a  point  in  M  dimensional  space  and  by  a  curve  in 
a  plane. 

III.  Linear  transformations.  The  transformations  to 
which  the  variables  in  an  w-ary  form  will  ordinarily  be  sub- 
jected are  the  following  linear  transformations  called  colline- 
ations  : 


1G 


THE   THEORY   OF   INVARIANTS 


(23) 


zn  =  Kz\  +  PrA  H h  ^X- 

In  algebraical  theories  the  only  restriction  to  which  these 
transformations  will  be  subjected  is  that  the  inverse  trans- 
formation shall  exist.  That  is,  that  it  be  possible  to  solve  for 
the  primed  variables  in  terms  of  the  un-primed  variables  (cf. 
(10)).  We  have  seen  in  Section  1,  V  (11),  and  VIII  (22) 
that  the  verification  of  a  covariant  and  indeed  the  very  exist- 
ence of  a  covariant  depends  upon  the  existence  of  this  inverse 
transformation. 

Theorem.  A  necessary  and  sufficient  condition  in  order 
that  the  inverse  of  (23)  may  exist  is  that  the  determinant  or 
modulus  of  the  transformation, 

\v   (iv  vv  .-.,  c 


M=  (Xfiv  •■•o-)  = 


\2,   /i2,   v 2<  •  •• ,  a. ^ 

K<   /V    *V   •")   <?n 


shall  be  different  from  zero. 

In  proof  of  this  theorem  we  observe  that  the  minor  of  any 
element,  as  of  jx^  of  M  equals Hence,  solving  for  a 

variable  as  x'v  we  obtain 

dM 


BM 


and    this  is   a  defined  result  in  all  instances   except   when 
M  =  0,  when  it  is  undefined.     Hence  we  must  have  M  =£  0. 

IV.  A  theorem  on  the  transformed  polynomial.      Let  /'be  a 
polynomial  in  xv  x2  of  order  m, 

f(xv  x2}  =  aQxf  +  ma^~xx^  +( ™  )a2x?~2xl  +  •••  +  amz%. 


THE  PRINCIPLES   OF   INVARIANT   THEORY       17 
Let/ be  transformed  into/'  by  T  (cf .  (3j)), 

f  =  a^+ma'jxf-1^  +  •••  +r*ja'rx'{H-rx''  -\ \-a'mx'™. 

We  now  prove  a  theorem  which  gives  a  short  method  of 
constructing  the  coefficients  a'r  in  terms  of  the  coefficients 

Theorem.     TJie  coefficients  a'r  of  the  transformed  form  /'  are 
given  by  the  formulas 


f  _\m  -r (       d 


a'  = 


d  \ 


[^dx  +  ^2ax~r  (Xr  Xa)  (r  =  °' "' m)"     (23l) 


In  proof  of  tliis  theorem  we  note  that  one  form  of/'  is 
/(A.-^  +  n-(x'v  \2x\  +  /a^)-  But  since/  is  homogeneous  this 
may  be  written 

/'  =  x'{\f{Xl  +  ^Jx'v  X2  +  AvIAi)- 

We  now  expand  the  right-hand  member  of  this  equality  by 
Taylor's  theorem,  regarding  x'%/x'x  as  a  parameter, 


/'=*;■ 


/(\1,\2)  +  :|^^;)/(X1,x2) 


where 


a\    f     a   ,      a 


/'  =f(\v  x^x^  + ...  h  y ^)r/(\v  \)*'rr4  + 


lA^axJ 

\m\   o\J 


18  THE   THEORY   OF   INVARIANTS 

Comparison  of  this  result  with  the  above  form  of/'  involving 
the  coefficients  a'r  gives  (23j). 

An  illustration  of  this  result  may  be  obtained  from  (5). 
Here  m  =  2,  and 

a'0  =  a0Xf  +  2  ^W  +  a2\l=f(\v  X2)  =/0, 

ai  =  a0\1/A1+a1(\1/*2+\2/i1)  +  a2\2/*2=-(/i—)/(\1,  \2),     (24) 


2Vd\J' 


1/     5  \2 
a2  =  a0nj  +  2  aj/ij^  +  a2/j%  =  -[fi—j  f(Xv  \2). 

V.  A  group  of  transformations.      If  we  combine  two  trans- 
formations, as  I7  and 

mi  .  x\  =  %\x\  "T"  Vi&zi 
X2  =  s2"^l     '     7?2"^2' 


there  results 

rpmi  .  #1  =  (Xl£l  +  Pl&M'  +  (\Vl  +  ^2)4'' 
'  *2  =  (X2?l  +  t*2%2)Xl  +  i\V  1+  /V?2  )4'« 

This  is  again  a  linear  transformation  and  is  called  the  prod- 
uct of  rand  T'.  If  now  we  consider  \v  A2,  /xv  fx.2  in  T  to 
be  independent  continuous  variables  assuming,  say,  all  real 
values,  then  the  number  of  linear  transformations  is  infinite, 
i.e.  they  form  an  infinite  set,  but  such  that  the  product  of  any 
two  transformations  of  the  set  is  a  third  transformation  of 
the  set.  Such  a  set  of  transformations  is  said  to  form  a 
group.  The  complete  abstract  definition  of  a  group  is  the 
following  : 

Given  any  set  of  distinct  operations  T,  T\  T'\  •••,  finite  or 
infinite  in  number  and  such  that : 

(a)  The  result  of  performing  successively  any  two  opera- 
tions of  the  set  is  another  definite  operation  of  the  set  which 
depends  only  upon  the  component  operations  and  the  sequence 
in  which  they  are  carried  out : 

(/?)  The  inverse  of  every  operation  T  exists  in  the  set ; 


THE   PRINCIPLES   OF   INVARIANT   THEORY       19 

that  is,  another  operation  T~l  such  that  TT1  is  the  identity 
or  an  operation  which  produces  no  effect. 

This  set  of  operations  then  forms  a  group. 

The  set  described  above  therefore  forms  an  infinite  group. 
If  the  transformations  of  this  set  have  only  integral  coeffi- 
cients consisting  of  the  positive  residues  of  a  prime  number 
p,  it  will  consist  of  only  a  finite  number  of  operations  and  so 
will  form  a  finite  group. 

VI.  The  induced  group.  The  equalities  (24)  constitute  a 
set  of  linear  transformations  on  the  variables  a0,  av  a2.  Like- 
wise in  the  case  of  formulas  (23x).  These  transformations 
are  said  to  be  induced  by  the  transformations  T.  If  T  carries 
/into/'  and  T'  carries/'  into/",  then 


\    3f 

I  m  —  r 


(r  =  0,  1,  ..-,  m). 

This  is  a  set  of  linear  transformations  connecting  the  a"r 
directly  with  a0,  •••,  am.  The  transformations  are  induced 
by  applying  T,  T'  in  succession  to/  Now  the  induced  trans- 
formations (23j)  form  a  group ;  for  the  transformations  in- 
duced by  applying  T  and  T'  in  succession  is  identical  with 
the  transformation  induced  by  the  product  TT' .  This  is 
capable  of  formal  proof.  For  by  (23x)  the  result  of  trans- 
forming/by TT'  is 

\m  —  r 
4  =  H=  =  A'/CX^  +  /*!&,  Xa^  +  /*2|2), 

\m 

where 

A  =  (X1t/1  +  /Ajife)  —  +  (X^!  +  fj,2vd 


3(X1|1  +  ^2)       "  "  J    3(X2f1  +  ^2) 


20  THE   THEORY   OF  INVARIANTS 

But 

*K\tityi&        all 

Hence 

and  by  the  method  of  (IV)  combined  with  this  value  of  A 


"§XlK)'f^-^r%- 


u      \m—r 
ar  ~     \m     V  dPJ  ^\s 

But  this  is  identical  with  (SIj).  Hence  the  induced  trans- 
formations form  a  group,  as  stated.  This  group  will  be 
called  the  induced  group. 

Definition.  A  quantic  or  form,  as  for  instance  a  binary 
cubic  /,  is  a  function  of  two  distinct  sets  of  variables,  e.g. 
the  variables  xv  x2,  and  the  coefficients  «0,  •••,  a3.  It  is  thus 
quaternary  in  the  coefficients  and  binary  in  the  variables 
xv  xv  We  call  it  a  quaternary-binary  function.  In  gen- 
eral, if  a  function  F  is  homogeneous. ana  of  degree  i  in  one 
set  of  variables  and  of  order  coin  a  second  set,  and  if  the  first 
set  contains  m  variables  and  the  second  set  n,  then  F  is  said 
to  be  an  w-ary-w-ary  function  of  degree-order  (i,  &>).  If  the 
first  set  of  variables  is  a0,  •••,  am,  and  the  second  2^,  •••,  xn, 
we  frequently  employ  the  notation 

F=(a0,  ...,  am)\.rv  ■■■.  xny. 

VII.  Cogrediency.  In  many  invariant  theory  problems 
two  sets  of  variables  are  brought  under  consideration  simul- 


THE    PRINCIPLES   OE   INVARIANT   THEORY       21 

taneously.  If  these  sets  (xv  xv  •  ••,  xn),  Qyv  yv  •  •-,  yn)  are 
subject  to  the  same  scheme  of  transformations,  as  (23),  they 
are  said  to  be  cogredient  sets  of  variables. 

As  an  illustration  of  cogredient   sets  we    first    take   the 
modular  binary  transformations, 

where  the  coefficients  X,  /u,  are  integers  reduced  modulo  ^>  as 
in  Section  1,  VIII.  We  can  prove  that  with  reference  to 
Tp  the  quantities  £f,  zf,   are  cogredient  to  xv  x%.     For  all 

binomial  numbers  f  A  J,  where  £>  is  a  prime,  are  divisible  by 

p  except  ( jj  J  and  (     )•     Hence,  raising  the  equations  of  Tp  to 

the  ^th  power,  we  have 

x\  =  Xfa^  +  ftfasgf,  zg  =  Xfz;^  +  A*!4P  (mod  iO- 

But  by  Fermat's  theorem, 

X?  =  X*  /*?  =  ^  (mod  />)     (t  =1,2). 
Therefore 

x\  =  \xx'l  +  /a^,  4'  =  \x'f  +  AV^sf' 

and  the  cogrediency  of  x\,  x\  with  xv  x2  under  Tp  is  proved. 

VIII.   Theorem.       The    roots    (r«>,    r2*>),     (rf ,     r22>),    •••, 
(j(m)^  r(m)^  0j?  a  Hnary  form 

f=aQx'{1  +  malx'{l-1x2+  ■■■  +  amx%, 
are  cogredient  to  the  variables. 
To  prove  this  we  write 

/=  (4%2  -  tfx^Cvfxi  ~  rfx^)  -  (r^n)xl  -  r<"!)z2), 
and  transform  /  by  T.     There  results 

m  .  , 

/'  =  n  [(4%  -  *•<%>£  +  (ffv,  -  ri'V3)4]. 

Therefore 

rj«>  =  r£%  -  rj%  ;  rf  =  -  (rgVi  -  r|« /*2). 


22  THE   THEORY   OF   INVARIANTS 

Solving  these  we  have 

(X/*)»f>  =  X/jW  +  ^w, 

(V*)rf  =  Viw  +  *Viw. 
Thus   the  r's  undergo   the   same   transformation  as  the  x's 
(save  for  a  common  multiplier  (X/*)),  and  hence  are  cogredi- 
ent  to  xv  xv  as  stated. 

IX.  Fundamental  postulate.  We  may  state  as  a  funda- 
mental postulate  of  the  invariant  theory  of  qualities  subject 
to  linear  transformations  the  following :  Any  covariant  of  a 
quantic  or  system  of  qualities,  i.e.  any  invariant  formation 
containing  the  variables  xv  xv  •••  will  keep  its  invariant 
property  unaffected  when  the  set  of  elements  xv  xv  •••  is 
replaced  by  any  cogredient  set. 

This  postulate  asserts,  in  effect,  that  the  notation  for  the 
variables  may  be  changed  in  an  invariant  formation  pro- 
vided the  elements  introduced  in  place  of  the  old  variables 
are  subject  to  the  same  transformation  as  the  old   variables. 

Since  invariants  may  often  be  regarded  as  special  cases 
of  covariants,  it  is  desirable  to  have  a  term  which  includes 
both  types  of  invariant  formations.  We  shall  employ  the 
word  concomitant  in  this  connection. 

Binary  Concomitants 

Since  many  chapters  of  this  book  treat  mainly  the  con- 
comitants of  binary  forms,  we  now  introduce  several  defini- 
tions which  appertain  in  the  first  instance  to  the  binary 
case. 

X.  Empirical  definition.     Let 

/=  a0a%  +  mojaf-1^  +  §  m(m  -  l)a2x\l~2xl  +  —  +  ama%, 

be  a  binary  form  of  order  m.     Suppose  /  is  transformed  by 
T  into 

/'  =  «;/«  +  ma[x[m-%  +  •••  +<4ro. 


THE   PRINCIPLES   OF   INVARIANT   THEORY       23 

We  construct  a  polynomial  (/>  in  the  variables  and  coeffi- 
cients of  /.  If  this  function  <£  is  such  that  it  needs  at  most 
to  be  multiplied  by  a  power  of  the  determinant  or  modulus 
of  the  transformation  (A.//.),  to  be  made  equal  to  the  same 
function  of  the  variables  and  coefficients  of  /',  then  0  is  a 
concomitant  of  /  under  T.  If  the  order  of  (f>  in  the  vari- 
ables xv  #2  is  zero,  <f>  is  an  invariant.  Otherwise  it  is  a  co- 
variant.  An  example  is  the  discriminant  of  the  binary 
quadratic,  in  Paragraph  III  of  Section  1. 

If  cf>  is  a  similar  invariant  formation  of  the  coefficients 
of  two  or  more  binary  forms  and  of  the  variables  xv  x2,  it  is 
called  a  simultaneous v  concomitant.  Illustrations  are  h  in 
Paragraph  IV  of  Section  1,  and  the  simultaneous  covariant 
C  in  Paragraph  V  of  Section  1. 

We  may  express  the  fact  of  the  invariancy  of  <f>  in  all 
these  cases  by  an  equation 

f =(\fiy<f>, 

in  which  <£'  is  understood  to  mean  the  same  function  of  the 
coefficients  a'0,  a'v  •••,  and  of  x'v  x'2  that  <f>  is  of  a0,  av  •••,  and 
xv  xv     Or  we  may  write  more  explicitly 

<}>(a'0,  a'v  •••;  x'v  x'%)  =  (\f*y4>  (a0,  av  •••;  xv  z2).        (25) 

We  need  only  to  replace  T  by  (23)  and  (X/m)  by  M  = 
(X/jl  ■■■  <r)  in  the  above  to  obtain  an  empirical 'definition  of  a 
concomitant  of  an  w-ary  form  /  under  (23).  The  corre- 
sponding equation  showing  the  concomitant  relation  is 

</>(«';  x'v  x'v  .-.,  x'n)  =  Mk(j>(a;  xv  xv  •  •,  xn).  (26) 

An  equation  such  as  (25)  will  be  called  the  invariant  rela- 
tion corresponding  to  the  invariant  <f>. 

XI.  Analytical  definition.*  We  shall  give  a  proof  in 
Chapter  II  that  no  essential  particularization  of  the  above 

*  The  idea  of  an  analytical  definition  of  invariants  is  due  to  Cayley.  Intro- 
ductory Memoir  upon  Quantics.     Works,  Vol.  II. 


24  THE    THEORY    OF   INVARIANTS 

definition  of  an  invariant  <f>  of  a  binary  form  /  is  imposed  by 
assuming  that  </>  is  homogeneous  both  in  the  a's  and  in  the 
x's.  Assuming  this,  we  define  a  concomitant  (f)  of  /  as 
follows  : 

(1)  Let  <£  be  a  function  of  the  coefficients  and  variables 
of/,  and  <j)'  the  same  function  of  the  coefficients  and  varia- 
bles of/'.     Assume  that  it  is  a  function  such  that 

^d^  2d\  1dfl1  2dfl2 

(2)  Assume  that  </>'  is  homogeneous  in  the  sets  \v  \2  ; 
fiv  /*2,  and  of  order  k  in  each. 

Then  <f>  is  called  a  concomitant  of/. 

We  proceed  to  prove  that  this  definition  is  equivalent  to 
the  empirical  definition  above. 

Since  </>'  is  homogeneous  in  the  way  stated,  we  have  by 
Euler's  theorem  and  (1)  above 

where  k  is  the  order  of  <f>'  in  \v  X2.     Solving  these, 


Hence 


oXj  oA2 

Separating  the  variables  and  integrating  we  have 

4>'       (V) 

where  (7  is  the  constant  of  integration.     To  determine  (7, 
let  T  be  particularized  to 


THE   PRINCIPLES    OF    INVARIANT   THEORY       25 

Then  a[  =  at (i  =  0,  1,  2,  •••,  ra),  and  </>'  =  <£.      Also  (Xyti)  =  1. 
Hence  by  substitution 

and  this  is  the  same  as  (25).     If  we  proceed  from 


x£>'-0'  ("£>'  =**'• 


we  arrive  at  the  same  result.  Hence  the  two  definitions  are 
equivalent. 

XII.    Annihilators.     We  shall  now  need  to  refer  back  to 
Paragraph  IV   (23x)  and  Section  1  (10)  and  observe  that 

("£K=(w-r)<+"  ("iX=0'  ("!iH-— '■ <29) 

Hence   the   operator  (/a —  )  applied    to  <£',    regarded   as   a 

function  of  \v  X2,  /t*j,  nv  has  precisely  the  same  effect  as 
some  other  linear  differential  operator  involving  only 
a[  (i  =  0,  •••,  m)  and  x'v  x'v  which  would  have  the  effect 
(29)  when  applied  to  <£'  regarded  as  a  function  of  a'^  x\, 
x'2  alone.  Such  an  operator  exists.  In  fact  we  can  see  by 
empirical  considerations  that 

rv        /d  i    &    i    s         n   /    3  •        5  ,     d 

0  -x1-=ma1-+(m-l)a2-+.--  +am^--x1-i 

(290 
is  such  an  operator.  We  can  also  derive  this  operator  by  an 
easy  analytical  procedure.     For, 


^  d\)V       da'0\     d\J^  da',  V    d\)  da'm\ 


or,  by  (29) 

(°'-^>'=°- 

26  THE   THEORY    OF    INVARIANTS 

III  the  same  manner  we  can  derive  from  f  X  —  ]<£'  =  0, 

Tlie  operators  (29x),  (292)  are  called  annihilators  (Sylvester). 
Since  <f>  is  the  same  function  of  aft  a^,  #2,  that  <£'  is  of  a',  »j, 
rr2,  we  have,  by  dropping  primes,  the  result  : 

Theorem.  A  set  of  necessary  and  sufficient  conditions  that 
a  homogeneous  function,  <f>,  of  the  coefficients  and  variables  of  a 
binary  form  f  should  be  a  concomitant  is 

In  the  case  of  invariants  these  conditions  reduce  to  Ocf>  =  0, 
Q<f>  =  0.  These  operators  are  here  written  again,  for  refer- 
ence, and  in  the  un-primed  variables: 

0  =  ma1—  +  (m  —  l)a2  —  -\ \-am- , 

da0  dax  Bam_x 

«-»  d        n        &  d 

il  =  a0— -+2a   — H h»w,H- — 

oax  l  da2  dam 

A  simple  illustration  is  obtainable  in  connection  with  the 
invariant 

Dl  =  a0a2  —  a\  (§  1,  III). 
Here  m  =  2  : 

D,D1  =  —  2  a0aj  -f-  2  aQax  =  0,  0DX  =  2  tf  ^  —  2  a^  =  0. 


It  will  be   noted  that  this  method  furnishes  a  convenient 
means  of  checking  the  work  of  computing  any  invariant. 


THE   PRINCIPLES   OF   INVARIANT   THEORY       27 


SECTION   3.     SPECIAL   INVARIANT   FORMATIONS 

We  now  prove  the  invariancy  of  certain  types  of  functions 
of  frequent  occurrence  in  the  algebraic  theory  of  quantics. 

I.   Jacobians.     Let  fvf2,  •••,/„  be  n  homogeneous  forms  in  n 
variables  xv  x2,  •••,  xn.     The  determinant, 


J= 


J2x{i       Jlx£        "'•>      J'2xn 
J  nXji       J  nx£         '  "l       J  nxn 


(30) 


in  which  /._  = -^,   etc.,  is  the  functional  determinant,  or 

Jacobian  of  the  n  forms.     We  prove  that  J  is  invariant  when 
the  forms /,-  are  transformed  by  (23),  i.e.  by 

Xi  =  \iX[  +  ^4  +  •  •  •  +  (TiX'n       (i  =  1,  2,   •  •  -,  71) .  (31) 

To  do  this  we  construct  the  Jacobian  J'  of  the  transformed 
quantic/j.     We  haveJSo&-f6i~), 


dx'2      dxx  dx'2      dx2  dx'2 


+ 


dxn  dx2 


But  by  virtue  of  the  transformations  (31)  we  have  in  all 
cases,  identically, 

/;=/,     (y  =  l,  2,  ...,n).  (32) 

Hence 


dx: 


dx, 


^2feH 


■•  +  *&.  (33) 

d.r„ 


1  "-2 

and  we  obtain  similar  formulas  for  the  derivatives  of  f'j  with 
respect  to  the  other  variables.     Therefore 


J'  = 


*l/l*l  +  \*/lsi  + r-Wutf  /"*i/i*,-+/*s/uiH Mn/lV 


*-l/«x  ,  +  W**  H H  X„/*V   /*l/nil  +  (hfnxt  H h  /*n/nx„i ' 


28 


THE   THEORY   OF   INVARIANTS 


But  this  form  of  J'  corresponds  exactly  with  the  formula 
for  the  product  of  two  nth.  order  determinants,  one  of  which 
is  J"  and  the  other  the  modulus  M.     Hence 

j'  =  (X/4-.o-y; 

and  J"  is  a  concomitant.  It  will  be  observed  that  the  co- 
variant  0  in  Paragraph  V  of  Section  1  is  the  Jacobian  of  / 
and  <f>. 

II.  Hessians.     If/ is  an  w-ary  form,  the  determinant 


H= 


Zl.Il1  J  x,_xS 


'1  J xlxn 
'i  J  x2xn 


J xnx£  J xnx£    '"ijxnxn 


(34) 


is  called  the  Hessian  of  /.  That  H  possesses  the  invariant 
property  we  may  prove  as  follows :  Multiply  H  by  M = 
(X/jlv  •••  <r),  and  make  use  of  (33).     This  gives 

d_df_     d    df  d    df 

dx\  dxx  dx'2  da^ '  dx'n  dx-^ 

d_  df_  J^cf_  J_df 

dx\  dx2  dx'2  dx2  '  dx'„  dx2 


H= 


^-n  Pn 


dx\  dxn    dx'2  dxn 
Replacing/ by/'  as  in  (32)  and  writing 

d    df        d    df 
ox\  dx-^       dxl  dx\ 

we  have,  after  multiplying  again  by  M, 


d    df 
dxL  dx„ 


Mm= 


p . .  f f . . 

J     XxXf  J    XnX^  1  J  xnx1 

J    x,i2*  . '  z„x2'  )  J  xnxt 

J  x,x„"   .'-r2J„*    "')J  xnxn 


THE   PRINCIPLES   OF   INVARIANT   THEORY       29 

that  is  to  say, 

and  i?is  a  concomitant  of/. 

It  is  customary,  and  avoids  extraneous  numerical  factors,  to 
define  the  Hessian  as  the  above  determinant  divided  by  lflnn 
x  (ffl  —  1)".     Thus  the  Hessian  covariant  of  the  binary  cubic 

LUx  111  j%  3     i     Q  2  i     Q  2    i  S 


is  *  A  =  2 


(35) 


=  2(a0a2  —  a^)xf  +  2(a0a3  —  a^i^)x-p^  +  2(a1a3  —  a\)x\. 

III.  Binary  resultants.      Let/,  $  be  two  binary  forms  of 
respective  orders,  w,  w ; 

/  =  a^  +  ma^""1^  -f-  •••  +  ama;^  =  H{r(^)x1  —  r^a^), 

0  =  bQx\  +  wi^-^ij  H h  ^2n  =  n(4/,^1  -  s(f  .r2). 

It  will  be  well  known  to  students  of  the  higher  algebra 
that  the  following  symmetric  function  of  the  roots  (r['\  r|°), 
(s[j\  s^)*1  -^(f*  $)  *s  ca^et^  the  resultant  of  /  and  <£.  Its 
vanishing  is  a  necessary  and  sufficient  condition  in  or  dot 
that /and  cj>  should  have  a  common  root. 

i2(  /,  0)  =  n  n  Of  ^  -  rfsp ) .  (36) 

j=iv=l 

To  prove  that  R  is  a  simultaneous  invariant  of  /  and  <£.  it 
will  be  sufficient  to  recall  that  the  roots  (rv  r2),  (sr  s2)  are 
cogredient  to  xv  x2.  Hence  when  /,  (f>  are  each  transformed 
by  T,  R  undergoes  the  transformation 

(\fi)s^  =  X/^  +  w'W,  (Xfi)s^  =  Vi y>  +  WP*,  *k> 

*  Throughout  this  book  the  notation  for  particular  algebraical  concomitants  is 
that  of  Clebsch. 


30 


THE   THEORY   OF   INVARIANTS 


in  which,  owing  to  homogeneity  the  factors  (X/x)  on  the  left 
may  be  disregarded.     But  under  these  substitutions, 

Hence 

i2'(/',^')  =  (V)mnS(/,<#>), 

which  proves  the  invariancy  of  the  resultant. 

The  most  familiar  and  elegant  method  of  expressing  the 
resultant  of  two  forms/,  cf>  in  terms  of  the  coefficients  of  the 
forms  is  by  Sylvester's  dialytic  method  of  elimination.  We 
multiply/  by  the  n  quantities  rrp1,  x1~2z2,  "'■>  a^-1  in  succes- 
sion, and  obtain 


a^+n~l     +  ma^™ +n~2x2  + 


anx 


m+n—2* 


+  mam_lx,{-ixn2l     +amx\-*x%+\  (37) 


Likewise  if  we  multiply  <£  by  the  succession  #"i_1,  a^~2xv  •  ••, 
ajjj*-1,  we  have  the  array 

J^4-"-1  +  nblx,[l+n~2x2  +  •••  +  bjf-hQ, 

Vi^1-1  +  ...  +  n5n_1r14'+n"2  +  &„^+n-1.  (38) 

The  eliminant  of  these  two  arrays  is  the  resultant  of/ and  <£, 
viz. 


R{M)= 


A  particular  case  of  a  resultant  is  shown  in  the  next  para- 
graph. The  degree  of  i2(/,</>)  in  the  coefficients  of  the  two 
forms  is  evidently  m  +  n. 


THE   PRINCIPLES   OF   INVARIANT   THEORY       31 

IV.  Discriminant  of  a  binary  form.  The  discriminant  D 
of  a  binary  form  /  is  that  function  of  its  coefficients  which 
when  equated  to  zero  furnishes  a  necessary  and  sufficient 
condition  ia  order  that/=  0  may  have  a  double  root.     Let 

f=f(xv  xi)  =  VT  +  wwhaf-1^  +  •••  +  amx'£, 

and  let  fXi(xv  x2)  =  ~-,  fx,(xv  xi)=a'     Tnen'  as  is  wel1 

df 
known,  a  common  root  of  /=  0,  -^—  =  0  is  a  double  root  of 

bx\ 

/=  0  and  conversely.     Also 

_,/    ,  df\      df 

hence    a   double   root   of  f=0   is  a  common  root  of  /=  0, 

—  =  0,  —  =  0,  and  conversely ;  or  D  is  equal  either  to  the 
dx1  dx2 

eliminant  of /and  -^-,  or  to  that  of/ and  — —  •     Let  the  roots 
dz1  dz2 

of  fx  (xv  x2)=0   be    (4°,  4*0(*  =  1)  "••>  m  —  1)'  those    of 

fXi(xv  x2)  =  0,  (tp,  tg>)(i  =  1,  ••-,  m  -  1),  and  those  of /=  0 

be  (r{"\  r2j))(j  =  1,  2,  •••,  m).     Then 

4o^=/(4{)'  tfOJW  42))---/(4w_1)'  sf_10> 
KP^fQP*  W(f}>  <f)  -/(^-^  e_10- 

Now  Of(xv  x2)=  xx^-,  Df(xv  ^2^=-r2^"'  where    0   and 
H  are  the  annihilators  of  Section  2,  XII.     Hence 

ai>= 24114^1),  tg))/(«^>,  4*)  .../«-i\  4-10=0. 

Thus  the  discriminant  satisfies  the  two  differential  equations 

OB  =  0,  £ID  =  0  and  is  an  invariant.    Its  degree  is  2£m  —  1). 

An  example  of    a  discriminant  is   the    following  for    the 

binary  cubic  /,  taken  as  the  resultant  of  -^-,  -*— : 

ox-l    ax2 


32 


THE   THEORY   OF   INVARIANTS 


-IR  = 


2ax 

a% 

0 

a0 

2ax 

a2 

2a2 

aB 

0 

ax 

2a2 

H 

(39) 


V. 


a0 
0 

0 
Universal  covariants.      Corresponding  to  a  given  group 


d  = 


=  (\/x)(:ry).        (40) 


of  linear  transformations  there  is  a  class  of  invariant  forma- 
tions which  involve  the  variables  only.  These  are  called 
universal  covariants  of  the  group.  If  the  group  is  the 
infinite  group  generated  by  the  transformations  T  in  the 
binary  case,  a  universal  covariant  is 

where  (j/)  is  cogredient  to  (x).     This  follows  from 
\x[  +  ixxx'2,  \2x\  +  (m2z'2  I 

If  the  group  is  the  finite  group  modulo  p,  given  by  the  trans- 
formations Tp,  then  since  xf,  x%  are  cogredient  to  xv  xv  we 
have  immediately,  from  the  above  result  for  d,  the  fact  that 
L  =  x\x%  -  xxx\  (41) 

is  a  universal  covariant  of  this  modular  group.* 

Another  group  of  linear  transformations,  which  is  of  con- 
sequence in  geometry,  is  given  by  the  well-known  trans- 
formations of  coordinate  axes  from  a  pair  inclined  at  an 
angle  co  to  a  pair  inclined  at  an  angle  co'  =  /3  —  a,*viz. 


x,  = 


sin  (co  —  a)    ,    ,  sin  (ft)  —  /3) 


sin  co 
sin  a 


A  + 


sin  co 


^2' 


,    .  sin/3 

•ri  +  - 


2* 


sin  co  sin  co 

Under  this  group  the  quadratic, 

rrf  +  2  xxx2  cos  co  +  x\\ 
is  a  universal  covariant. f 

*  Dickson,  Transactions  Amer.  Math.  Society,  vol.  12  (1911). 
t  Study,  Leipz.  Ber.  vol.  40  (1897). 


(42) 


(43) 


CHAPTER   II 

PROPERTIES  OF  INVARIANTS 

SECTION   1.    HOMOGENEITY  OF   A   BINARY  CONCOMITANT 

I.    Homogeneity.     A  binary  form  of  order  m 

f=  a0xf  +  ma1zf~1x2  -\ h  Qmx"2\ 

is  an  (m  +  l)-ary-binary  function  of  degree-order  (1,  mi). 
A  concomitant  of  /  is  an  (w  +  l)-ary-binary  function  of  de- 
gree-order (i,  (o).  Thus  the  Hessian  of  the  binary  cubic 
(Chap.  I,  §  3,  II), 

A  =  2(a0a2  —  af)x%  +  2(a0a3  —  axa2)xxx2  +  2(a1a3  —  a|)x|,  (44) 

is  a  quaternary-binary  function  of  degree-order  (2,  2). 
Likewise  /+  A  is  quaternary-binary  of  degree-order  (2,  *£), 
but  non-homogeneous. 

An  invariant  function  of  degree-order  (i,  0)  is  an  invariant 
of  /.  If  the  degree-order  is  (0,  «),  the  function  is  a  universal 
covariant  (Chap.  I,  §  3,  V).  Thus  a0a2  —  a\  of  degree-order 
(2,  0)  is  an  invariant  of  the  binary  quadratic  under  T, 
whereas  x\x2  —  xxx\  of  degree-order  (0,  p  +  1)  is  a  universal 
modular  covariant  of  Tp. 

Theorem.  If  C=  (a0,  a^  •••,  ani)\xv  x^)m  is  a  concomitant 
of  /=(a0,  ■••,  am~)(xv  x2)m,  its  theory  as  an  invariant  function 
loses  no  generality  if  toe  assume  that  it  is  homogeneous  both  as 
regards  the  variables  xv  x2  and  the  variables  a0,  •••,  am. 

Assume  for  instance  that  it  is  non-homogeneous  as  tozj,  x2. 
Then  it  must  equal  a  sum  of  functions  which  are  separately 
homogeneous  in  xv  x2.     Suppose 

33 


34  THE   THEORY   OF   INVARIANTS 

■V   . 

where  (7;-  =  (a0,  aj,  •••,  am)i'(xv  x2)tcj(j  =  1,  2,  ••-,  s')  i' -^i. 
Suppose  now  that  we  wish  to  verify  the  covariancy  of  C, 
directly.     We  will  have 

a  =  «,  a[, ...,  o*(^,  4)«  =  (\pyo,        (45) 

in  which  relation  we  have  an  identity  if  a\  is  expressed  as  the 
appropriate  linear  expression  in  a0,  ••.,  am  and  the  ^  as  the 
linear  expression  in  xv  xv  of  Chapter  I,  Section  1  (10).  But 
we  can  have 

identically  in  xv  xv  only  provided 

Hence  (7,-  is  itself  a  concomitant,  and  since  it  is  homogeneous 
as  to  Zj,  a:2,  no  generality  will  be  lost  by  assuming  all  invariant 
functions  C  homogeneous  in  xv  x2. 

Next  assume  C  to  be  homogeneous  in  xv  x2  but  not  in  the 
variables  «0,  av  •••,  am.     Then 

c=r1  +  r2  +  ...+r(T, 

where  T,  is  homogeneous  both  in  the  a's  and  in  the  x's.  Then 
the  above  process  of  verification  leads  to  the  fact  that 

r;=0)*r„ 

and  hence  C  may  be  assumed  homogeneous  both  as  to  the  as 
and  the  x's ;  which  was  to  be  proved.  The  proof  applies 
equally  well  to  the  cases  of  invariants,  covariants,  and  uni- 
versal covariants. 

SECTION  2.     INDEX,   ORDER,   DEGREE,   WEIGHT 

In  a  covariant  relation  such  as  (45)  above,  k,  the  power  of 
the  modulus  in  the  relation,  shall  be  called  the  index  of  the 
concomitant.  The  numbers  i,  <o  are  respectively  the  degree 
and  the  order  of  C.  ■ 


PROPERTIES   OF   INVARIANTS  35 

I.  Definition.  Let  t  =  a%a\a'2  •  •  •  a^x^x^~^  be  any  monomial 
expression  in  the  coefficients  and  variables  of  a  binary  m-ief 
The  degree  of  r  is  of  course  i  =p  4-  q  +  r  4  •••  +  v.  The 
number 

w  =  q+2r+3s-\ \-  mv  4  fi  (46) 

is  called  the  weight  of  t.  It  equals  the  sum  of  all  of  the  sub- 
scripts of  the  letters  forming  factors  of  r  excluding  the  factors 
x2.  Thus  a3  is  of  weight  3  ;  aQa\aA  of  weight  6 ;  a\a^c\x\  of 
weight  9.  Any  polynomial  whose  terms  are  of  the  type  t 
and  all  of  the  same  weight  is  said  to  be  an  isobaric  polynomial. 
We  can,  by  a  method  now  to  be  described,  prove  a  series  of 
facts  concerning  the  numbers  &>,  i,  k,  w. 

Consider  the  form  /  and  a  corresponding  concomitant 
relation 

C    =  (a0,  Oj,  •••,  am)l(^x^  x2y 

=  (V)fc(a0,  av  •••,  amy(xv  x2Y.      (47) 

This  relation  holds  true  when  /  is  transformed  by  any  linear 
transformation 

Xx  =  X.J2-J  -H  ^1^2' 
x1  =  ^2^  1  ~^~  ^lXT 

It  will,  therefore,  certainly  hold  true  when  /  is  transformed 
by  any  particular  case  of  T.  It  is  by  means  of  such  particu- 
lar transformations  that  a  number  of  facts  will  now  be  proved. 

II.  Theorem.      TJie   index   k,   order  a>,  and   degree  i  of  0 

satisfy  the  relation 

k  =  ±(im-a>}.  (48) 

And  this  relation  is  true  of  invariants,  i.e.  (48)  holds  true  when 
(o=0. 

To  prove  this  we  transform 

/=  a0x™  +  ma^xf-%  +  •••  +  aj%, 

by  the  following  special  case  of  T: 


36  THE   THEORY   OF   INVARIANTS 

The  modulus  is  now  X2,  and  a'j=  Xma,-  (J=  0,  •••,  rri).     Hence 
from  (47), 

(X"a0.  \mav  ....  Xmam)*(X-1z1.  X"1.^)" 

=  X2*(a0,  op  •-.,  0*0*11  *„)».      (49) 

But  the  concomitant   C  is  homogeneous.     Hence,  since  the 
degree-order  is  (£,  co ). 

X*»— (a0,  •-.  O*0*n  a^)"  =  X2*(a0,  ....  0*0*11  x2y. 

Hence 

2  k  =  im  —  co. 

III.   Theorem.     Every  concomitant   C  of  f  is  isobaric  and 
the  weight  is  given  by 

iv  =  \(im  +  <u),  (50) 

where  (z,  co)  w  £/?<>  degree-order  of  C.  and  m  the  order  of  f. 
The  relation  is  true  for  invariants,  i.e.  if  (0  =  0. 

In  proof  we  transform  /  by  the  special  transformation 

Then  the  modulus  is  X,  and  aj=  Va,-  0  =  0,  2.  •••.  in). 
Let 

be  any  term  of   C  and  t'  the  corresponding  term  of  C  the 
transformed  of  O  by  (51).     Then  by  (-±7). 

Thus 

ic  —  co  =  k  =  l(  im  —  co). 
or 

ic  =  J(ww  +  O)  ). 

Corollary  1.      The    weight    of    an    invariant   equals  its 
index. 

w  =  Jc  =  \  im . 

Corollary  2.     The  degree-order  (i.  co)  of  a  concomitant 
C  cannot  consist  of   an  even   number  and  an  odd  number 


PROPERTIES   OF   INVARIANTS  37 

except  when  m  is  even.  Then  i  may  be  odd  and  co  even. 
But  if  m  is  even  co  cannot  be  odd. 

These  corollaries  follow  directly  from  (48),  (50). 

As  an  illustration,  if  O  is  the  Hessian  of  a  cubic,  (44),  we 

have 

t  =  2,  co  =  2,  m  =  3, 

W  =  i(2-3  +  2)  =  4, 

*  =  £(2  .  3  -  2)  =  2. 

These  facts  are  otherwise  evident  (cf.  (44),  and  Chap. 
I,  §3,  II). 

Corollary  3.     The  index  k  of  any  concomitant  of  /  is  a 
positive  integer. 
For  we  have 

w  —  co  =  k, 

and  evidently  the  integer  w  is  positive  and  co  4.  w. 

SECTION  3.     SIMULTANEOUS   CONCOMITANTS 

We  have  verified  the  invariancy  of  two  simultaneous 
concomitants.  These  are  the  bilinear  invariants  of  two 
quadratics  (Chap.  I,  §  1,  IV), 

rfr  =  a0.r'l  +  2  fl^a^a^  +  a2x|, 

4>  =  <V1  +  -  VV'2  +  hzv 
viz.  h?=a0b2  —  2  a-fil  +  a2bQ, 

and  the  Jacobian  C  of  |  and  cf>  (cf.  (8)).  For  another 
illustration  we  may  introduce  the  Jacobian  of  cf>  and  the 
Hessian,  A,  of  a  binary  cubic/.     This  is  (cf.  (44)) 

J*.  A  =  [&0(«oa3  -  a\al)-  ~  h\(.a«a2  -  a°V~\X'\ 

+  2  [b0(a1a3  -  a\)-  b2(a0a2  -  af)]^ 
+  [-  ^i(«i«3  -  aD  ~  h(ao(h  ~  a\ai  )>l 
and  it  may  be  verified  as  a  concomitant  of  c\>  and 
f=a03%+  -. 


38  THE   THEORY   OF   INVARIANTS 

The  degree-order  of  J  is  (3,  2).  This  might  be  written 
(1  +  2,  2),  where  by  the  sum  1  +  2  we  indicate  that  J  is  of 
partial  degree  1  in  the  coefficients  of  the  first  form  $  and  of 
partial  degree  2  in  the  coefficients  of  the  second  form  /. 

I.  Theorem.  Let  f  (f>,  \^,  ...  be  a  set  of  binary  forms  of 
respective  orders  mv  w2,  m3,  •  ••.  Let  C  be  a  simultaneous 
concomitant  of  these  forms  of  degree-order 

(ix  +  i2  +  i3+  •  •-,  oo). 

Then  the  index  and  the  weight  of  C  are  connected  with  the 
numbers  m,  i,  oo  bg  the  relations 

k  =  \(^i1m1  —  &)),  (52) 

w  =  \{  '2.ilm1  +  to), 

and  these  relations  hold  true  for  invariants  (i.e.  when  oo  =  0). 

The  method  of  proof  is  similar  to  that  employed  in  the 
proofs  of  the  theorems  in  Section  2.  We  shall  prove  in 
detail  the  second  formula  only.      Let 

/=  a0x'^  +  -..,  $  =  Vi"2  +    ••,  ^  =  c^  +  •••,  .... 
Then  a  term  of  0  will  be  of  the  form 

r  =  a^a[m^  •••  bfrbfify  •••  x^xf*. 

Let  the  forms  be  transformed  by  xx  =  x'v  x2  =  \x'2.  Then  aj 
=  \jah  b)  =  \jbj,  ...  (J  =  0,  •-.,  ?w(),  and  if  r'  is  the  term  cor- 
responding to  t  in  the  transformed  of  O  by  this  particular 
transformation,  we  have 

T1  =  \rl+2s1+-+rt+2si+.~+lt.-o>T  _   ^*T_ 

Hence 

xv  —  oo  =  k  =  \{'2i1m1  —  &>), 

which  proves  the  theorem. 

We  have  for  the  three  simultaneous  concomitants  men- 
tioned above  ;  from  formulas  (52) 

h  C  J 

k  =  2  k  =  l  fc  =  3 

w =  2  w  =  3  w  =  5 


PROPERTIES   OF   INVARIANTS  39 

SECTION  4.   SYMMETRY.     FUNDAMENTAL   EXISTENCE 
THEOREM 

We  have  shown  that  the  binary  cubic  form  has  an  invari- 
ant, its  discriminant,  of  degree  4,  and  weight  6.  This  is 
(cf.  (39)) 

!  R  =  —  (aQa3  —  a^)2  +  -I(a0a2  —  «i)(«i«3  —  a\)- 

I.  Symmetry.  We  may  note  concerning  it  that  it  is 
unaltered  by  the  substitution  0v*3Kaia2)'  This  fact  is  a 
case  of  a  general  property  of  concomitants  of  a  binary  form 
of  order  m.  Let/=  a0x^  +  •••  ;  and  let  0  be  a  concomitant, 
the  invariant  relation  being 

C  =  «,  a'v  •••,  a'my(x'v  4)w  =  (\fiy(a0,    •  •,  amy(xv  x2y. 

Let  the  transformation  Toff  be  particularized  to 

The  modulus  is  —  1.     Then  a'j  =  am_;>  and 

a) 
C'  =  (am,  am_v  ••-,  aQy(xT  tc1)<-=(-l)A(a0,  •••,  amy(xv  x2).    (53) 

That  is  ;  any  concomitant  of  even  index  is  unchanged  when 
the  interchanges  Qa0am)(a1am_1)  •••  {xxx^)  are  made,  and  if 
the  index  be  odd,  the  concomitant  changes  only  in  sign. 
On  account  of  this  property  a  concomitant  of  odd  index  is 
called  a  skew  concomitant.  There  exist  no  skew  invariants 
for  forms  of  the  first  four  orders  1,  2,  3,  4.  Indeed  the 
simplest  skew  invariant  of  the  quintic  is  quite  complicated, 
it  being  of  degree  18  and  weight  45*  (Hermite).  The  sim- 
plest skew  covariant  of  a  lower  form  is  the  covariant  T  of  a 
quartic  of  (125)  (Chap.  IV,  §  1). 

We  shall  now  close  this  chapter  by  proving  a  theorem 
that  shows  that  tne  number  of  concomitants  of  a  form  is 
infinite.  We  state  this  fundamental  existence  theorem  of 
the  subject  as  follows  : 

*  Faa  di  Bruno,  Walter.    Theorie  der  Binaren  Formen,  p.  320. 


40  THE   THEORY   OF   INVARIANTS 

II.  Theorem.  Every  concomitant  K  of  a  covariant  Q  of  a 
binary  form  f is  a  concomitant  off 

That  this  theorem  establishes  the  existence  of  an  infinite 
number  of  concomitants  of/  is  clear.  In  fact  if/  is  a  binary 
quartic,  its  Hessian  covariant  H  (Chap.  I,  §  3)  is  also  a 
quartic.  The  Hessian  of  H  is  again  a  quartic,  and  is  a  con- 
comitant of  /  by  the  present  theorem.  Thus,  simply  by 
taking  successive  Hessians  we  can  obtain  an  infinite  number 
of  covariants  of  /,  all  of  the  fourth  order.  Similar  consider- 
ations hold  true  for  other  forms.  ■ 

In  proof  of  the  theorem  we  have 

/=a0,-  +  ..., 

(7=(a0,  ...,  amy(xv  x2y  =  cQxf  +  tae1aq-1x%  H , 

where  ci  is  of  degree  i  in  a0,  •••,  am. 

Now  let  /  be  transformed  by  T.  Then  we  can  show  that 
this  operation  induces  a  linear  transformation  of  (7,  and 
precisely  T.  In  other  words,  when/  is  transformed,  then 
C  is  transformed  by  the  same  transformation.  For  when  / 
is  transformed  into/',  -0  goes  into 

C'  =  (\ny(c0x»  +  (oc^-\r2  +  ...)•  ' 

But  when  C  is  transformed  directly  by  T,  it  goes  into  a  form  / 
which  equals  C  itself  by  virtue  of  the  equations  of  trans- 
formation. Hence  the  form  (7,  induced  by  transforming  /, 
is  identical  with  that  obtained  by  transforming  C  by  T 
directly,  save  for  the  factor  (\ix)k.  Thus  hx  transformation 
of  either  /  or  (7, 

c0'a^°  +  (oc'^-^  H =  (X/i)*c0ry  +  (o(\/jiyc1xf~1.r2  -j (54) 

is  an  equality  holding  true  by  virtue  of  the  equations  of 
transformation.  Now  an  invariant  relation  for  K  is  formed 
by  forming  an  invariant  function  from  the  coefficients  and 
variables  of  the  left-hand  side  of  (54)  and  placing  it  equal 
to  (Xfi)K  times  the  same  function  of  the  coefficients  and 
the  variables  of  the  right-hand  side, 


PROPERTIES    OF   INVARIANTS  41 

But  .ST'  is  homogeneous  and  of  degree-order  (t,  e).     Hence 

=  (VO**+"if. 

Now  cj  is  the  same  function  of  the  a'0,  •  ••,  aj„  that  <?7-  is  of  a0, 
•  ••,  am.  When  the  ons  and  c's  in  (55)  are  replaced  by 
their  values  in  terms  of  the  a's,  we  have 

where,  of  course,  [a0,  •••,  am]u(xv  x2)e  considered  as  a  func- 
tion, is  different  from  (a0,  •  ••,  am)h(xv  x2~)€.  But  (56)  is 
a  covariant  relation  for  a  covariant  of  f.  This  proves  the 
theorem./ 

The  proof  holds  true  mutatis  mutandis  for  concomitants  of 
an  w-ary  form  and  for  simultaneous  concomitants. 

The  index  of  iT  is 

p  =  i .  -1  ( im  —  to)  +  ^-(i&>  —  e) 

=  ]-(Mffl  —  e), 
and  its  weight, 

w  =  \{iim  +  e). 
Illustration.     If  /is  a  binary  cubic, 

then  its  Hessian, 

A  =  2[(rt0a2  —  a\)x\  +(a0ag  —  a^x^x^  +  (axaz  —  af)a|], 

is  a  covariant  of/.  The  Hessian  2 H  of  A  is  the  discrimi- 
nant of  A,  and  it  is  also  twice  the  discriminant  of/, 

2  i2  =  4[—  (rt0a3  —  a1a2)2  +  4(«0<z2  _  r(i)(aia3  —  a|)]- 


CHAPTER   III 

THE  PROCESSES   OF   INVARIANT  THEORY 
SECTION   1.     INVARIANT   OPERATORS 

We  have  proved  in  Chapter  II  that  the  system  of  invari- 
ants and  covariants  of  a  form  or  set  of  forms  is  infinite. 
But  up  to  the  present  we  have  demonstrated  no  methods 
whereby  the  members  of  such  a  system  may  be  found.  .The 
only  methods  of  this  nature  which  we  have  established  are 
those  given  in  Section  3  of  Chapter  I  on  special  invariant 
formations,  and  these  are  of  very  limited  application.  We 
shall  treat  in  this  chapter  the  standard  known  processes  for 
finding  the  most  important  concomitants  of  a  system  of 
quantics. 

I.  Polars.  In  Section  2  of  Chapter  I  some  use  was  made 
of  the  operations  Xx \-  X9  — ,  /xx f-  /li9 Such  opera- 

tors  may  be  extensively  employed  in  the  construction  of  in- 
variant formations.      They  are  called  polar  operators. 

Theorem.  Let  f  =  a^x'[l -\-  •••  be  an  n-ary  quantic  in  the 
variables  xv  •••,  xn,  and  cf>  a  concomitant  off,  the  correspond  in;  i 
invariant  relation  being 

£'=<X,  ■•  0*0*4.  •••,  <y 

=  (\fi  •-  o-)*o0,  .-0*Oi.  •••,  xny  =  3P<f>.  (57) 

Then  if  yv  yv  •••,  yn  are  cogredient  to  xv  .r2,  •••,  xn,  the  function 
is  a  concomitant  off 

42 


THE   PROCESSES   OF   INVARIANT   THEORY        43 
It  will  be  sufficient  to  prove  that 

the  theorem  will  then  follow  directly  by  the  definition  of  a 
covariant.     On  account  of  cogrediency  we  have 

xt  =  \ix[  +  ,iv  2  +  •••  +  o-fh  (59) 


Hence 


d      _  _3_  d^      j9^  d.r2  3   dxn 

bx\       dx1  d.r[       dx2  d.i\  dxn  dx\ 


ox2  dx±  d.r2  dxn 


Therefore 


3  3  3  a 

— 7  =  o"i h  cr2 1"  "'  +  °"n — " 

dxn  bxx  dx2  dxn 


3  n  rj 

#i^7+  •••  +  ^'v-'  =(Xi^i  +  W2  +  •••  +  °"i#Ot- 

•J 

+   ...  +(\nyl+ixny>+   ...  +  (jny'n)—- 

d  3 

dxx  dxn 

Hence  (58)  follows  immediately  when  we  operate  upon  (57) 

The  function  (y — )<£   is    called  the   first  polar   covariant  of 

4>,  or  simply  the  first  polar  of  (f).      It  is  convenient,  however, 
and  avoids    adventitious  numerical  factors,  to  define  as  the 

polar  of  cj>  the  expression   [y — )$  times  a  numerical  factor. 

V    dxj 


44 


THE   THEORY   OF   IX VARIANTS 


We  give  this  more  explicit  definition  in  connection  with 
polars  of  /  itself  without  loss  of  generality.  Let  /  be  of 
order  m.     Then 


\m 


d 


+  Vi 


d.r, 


J  =Jyri 


(61) 


the  right-hand  side  being  merely  an  abbreviation  of  the  left- 
hand  side,  is  called  the  rth  ?/-polar  of  /.  It  is  an  absolute 
covariant  of/ by  (60). 

For  illustration,  the  first  polars  of 

f  =  aQx-j  +  3  axx\x2  +  3  a^-^c\  +  azx\, 

9  =  a200X\  ~t~  *  al\0x\X,i  "I"  ^020^2  ~^~  ~  ^lOl^l2^  ~^  "  ^Oll3^"^  ~^~  a002•r3, 

are,  respectively, 

fy  =  (aoxl  +  Sa^jarjj  +  a2x^)y1  +  (axx\  +  2  a2xxx2  +  a#%)yv 

9y  =  (^200^1  +  ^no2^ "I"  ^ioi2^  )^i ~f~  v^no^i  "^"  aa&&r2  "+"  ^011^3)^2 

+  (rtjo!^  +  #  oil2^  +  ^02^3)^3- 

Also, 

//  =  (%y\  +  2  «li/l^2  +  a2i/|>-rl  +  («#!  +  2  a29llh  +  ^i)^- 
If  ^/  =  0  is  the  conic  C  of  the  adjoining  figure,  and  (j/)  = 

(jyv  yv  y^)  is  the  point  P,  then  </y  =  0  is  the  chord  of  contact 
AB,  and  is  called  the  polar  line  of  P 
and  the  conic.  If  P  is  within  the  conic, 
gy  =  0  joins  the  imaginary  points  of 
contact  of  the  tangents  to  C  from  P. 

We  now  restrict  the  discussion  in  the 
remainder  of  this  chapter  to  binary 
forms. 

We  note  that  if  the  variables  (?/)  be 
replaced  by  tJie  variables  (x)  in  any  polar 

of  a  form  f  the  result  is  f  itself  i.e.  the  original  polarized 

form.     This  follows   by   Euler's  theorem   on   homogeneous 

functions,  since 


*P 


^ 


_d_ 

dx, 


+  x2—]f=mf. 


(62) 


THE   PROCESSES   OF   INVARIANT   THEORY        45 

In  connection  with  the  theorem  on  the  transformed  form 
of  Chapter  I,  Section  2,  we  may  observe  that  the  coefficients 
of  the  transformed  form  are  given  by  the  polar  formulas 

a'0=f(XvX2)=fr  (63) 

The  rth  y-polar  of  f  is  a  doubly  binary  form  in  the  sets 
(jJv  #2)'  ixv  x%)  °f  degree-order  (r,  m  —  r).  We  may  how- 
ever polarize  f  a  number  of  times  as  to  (j/)  and  then  a  number 
of  times  as  to  another  cogredient  set  (z)  ; 


u 


\m  —  r\m  —  r  —  sf    3  y/    d  , 


This  result  is  a  function  of  three  cogredient  sets  (#),  (j/),  (z). 
Since  the  polar  operator  is  a  linear  differential  operator,  it 
is  evident  that  the  polar  of   a  sum  of  a  number  of  forms 
equals  the  sum  of  the  polars  of  these  forms, 

(/+  <t>  +  -V=A'  +  <#y+  •••• 

II.  The  polar  of  a  product.  We  now  develop  a  very  im- 
portant formula  giving  the  polar  of  the  product  of  two 
binary  forms  in  terms  of  polars  of  the  two  forms. 

If  F(xv  #2)  is  any  binary  form  in  xv  x2  of  order  M  and  (y) 
is  cogredient  to  (V),  we  have  by  Taylor's  theorem,  k  being 
any  parameter, 

JPOi  +  %r  x2  +  kVi) 

-*+(*)*.»+(?)*,*  +  -  +(?)***+  -•       w 

Let  F=f(xv  x2)<f)(xv  rr2),  the  product  of  two  binary  forms 
of  respective  orders  w,  w.  Then  the  rth  polar  of  this  prod- 
uct will  be  the  coefficient  of  kT  in  the  expansion  of 

f(xx  +  kyv  x2  +  %2)  x<f>(x1  +  kyv  x2  +  %2), 


46 


THE   THEORY   OF   INVARIANTS 


divided  by  (       .      ),  by  (65).     But  this  expansion  equals 
+  (oW2+  -+("W  +  -" 


Hence  by  direct  multiplication, 


r 


or 


f* 


Io)(^KHT)(-i)^1+- 


(66) 


This  is  the  required  formula. 

The  sum  of  the  coefficients  in  the  polar  of  a  product  is 
unity.  This  follows  from  the  fact  (cf.  (62))  that  if  (y) 
goes  into  (x)  in  the  polar  of  a  product  it  becomes  the  origi- 
nal polarized  form. 

An  illustration  of  formula  (66)  is  the  following : 

Let/=a0:rj  +  -•-,  cf>  =  b0xj+  ....     Then 
20 


/* 


'  ro)f^ + (t)  (iV^ + (i)  (IVa 


=  ifA  +  UrA  +  lf!A>- 


III.    Aronhold's  polars.     The  coefficients  of  the  transformed 
binary  form  are  given  by 

a'r  =./>(Ar  *sXr  =  °'  — i  m)- 

These  are  the  linear  transformations  of  the  induced  group 
(Chap.  I,  §  2).  Let  <f>  be  a  second  binary  form  of  the  same 
order  as,/, 

cf>  =  b0a^4-  >nbvv';>-\r2+  .... 


THE   PROCESSES   OF   INVARIANT   THEORY        47 
Let  cf>  be  transformed  by  ^into  cf>' .     Then 

Hence  the  set  60,  bv  •  ••,  bm  is  cogredient  to  the  set  aQ,  av  •  ••,  am, 
under  the  induced  group.  It  follows  immediately  by  the 
theory  of  Paragraph  I  that 

=  50t-+  •••  +bm—=[b^-).  (67) 

da0  dam     \  daj 

That  is,  [b  —  j  is  an  invariant  operation.      It  is  called  the 

Aronhold  operator  but  was  first  discovered  by  Boole  in  1841. 
Operated  upon  any  concomitant  of/  it  gives  a  simultaneous 
concomitant  of/  and  <f>.     If  m  =  2,  let 

1=  a0a2  —  a\. 
Then 

day         \     daQ  dax  da2 

This  is  h  (Chap.  I,  §  1).     Also 


b  —  )I=(b0- h^- \-b2-—)I  =  a0b2  —  2a1b1  +  a2b0. 


2(60J=4(5ofi2-5f), 


the  discriminant  of  </>.     In  general,  if  yjr  is  any  concomitant 
of/, 

then   (^J+'-M4^'*/'-^.?^    (68) 

are  concomitants  of /and  <£.     When  r  =  i,  the  concomitant  is 

The  other  concomitants  of  the  scries,  which  we  call  a  series 
of  Aronhold's  polars  of  -v/r,  are  said  to  be  intermediate  to  yfr 


48  THE   THEORY   OF   INVARIANTS 

and  x->  and  of  the  same  type  as  yjr.  The  theory  of  types  will 
be  referred  to  in  the  sequel. 

All  concomitants  of  a  series  of  Aronhold's  polars  have  the 
same  index  k. 

Thus  the  following  series  has  the  index  Jc  =  2,  as  may  be 
verified  b}~  applying  (52)  of  Section  3,  Chapter  II  to  each 
form  (/=  a^x\  +  •••  ;   <f>  =  b<p%  +  •••): 

H=  (a0a2  —  «?)•*'?  +  (#0^3  ~~  aia'i)x\x2  +  (.axa8  ~  a%)x% 
(b—]ff={a0b2—  2  a151+a2J0)a;j+(a06g+a360— aj^—  a^b^x^ 
V  +(a1b3-2a2b2+a8b1)xl 

l(t>£f&=  chh  -  h\>*i + c  V'3  -  hh>i** + (.hh  -  H)4- 

IV.  Modular  polars.  Under  the  group  Tp,  we  have  shown, 
x\,  x\  are  cogredient  to  xv  xv     Hence  the  polar  operation 

^^r  +  ^r*  (69) 

ox1  dx2 

applied  to  any  algebraic  form  /,  or  covariant  of  /,  gives  a 
formal  modular  concomitant  of/.     Thus  if 

/  =  a0x%  +  2  axx^x2  +  a2x\, 
then, 

1  S3/=  a0x\  +  a1(a%x2  +  xxx\ )  +  «2;4- 

This  is  a  covariant  of  /  modulo  3,  as  has  been  verified  in  Chap- 
ter I,  Section  1.  Under  the  induced  modular  group  a*,  of,  •••, 
afn  will  be  cogredient  to  «0,  av  •••,  am.  Hence  we  have  the 
modular  Aronhold  operator 

da0  dam 

If  m  =  2,  and 

D  =  a0a2  -  of, 

then  dpZ)  =  agog  -  2  of+1  +  a0a|  (mod.  p). 

This  is  a  formal  modular  invariant  modulo  p.     It  is  not  an 


THE   PROCESSES   OF   INVARIANT   THEORY        49 

algebraic  invariant ;   that  is,  not  invariantive  under  the  group 
generated  by  the  transformations  T. 
We  may  note  in  addition  that  the  line 

has  among  its  covariants  modulo  2,  the  line  and  the  conic 

d2l  =  af)x1  +  a\x2  +  a\xy 
82l  =  atfc\  +  axx\  +  a2x\. 

X.  Operators  derived  from  the  fundamental  postulate.  The 
fundamental  postulate  on  cogrediency  (Chap.  1.  §  -J  enables 
us  to  replace  the  variables  in  a  concomitant  by  any  set  of  ele- 
ments cogredient  to  the  variables,  without  disturbing  the 
property  of  invariance. 

Theorem.      Under  the  binary  transformations  T  the  differ- 
ential operators  - — ,  — —  are  cogredient  to  the  variables. 
dx2         bxx 

From  T  we  have 

dx'1         ldx1         lbx2 
bob 

Hence  (W)— -  =  X1^—  +  /iJ  —  — ), 

ofi,  los.,         \      Ox\J 

n      d     1    d        (     b  \ 

-(x^ox~r  ^ +**  rw 

This  proves  the  theorem. 

It  follows  that  if  <£='(a0.  ....  amy(xv  x2/"  is  any  invariant 
function,  i.e.  a  concomitant  of  a  binary  form/,  then 


ty^...,^_,__j 


70 


is  an  invariant  operator  (Boole).     If  this  operator  is  operated 

upon  any  covariant  of  /.   it  gives  a  concomitant  of  /,  and 


50  THE   THEORY   OF   INVARIANTS 

if  operated  upon  a  covariant  of  any  set  of  forms  </,  /i,  ...,  it 
gives  a  simultaneous  concomitant  of  /  and  the  set.  This 
process  is  a  remarkably  prolific  one  and  enables  us  to  construct 
a  great  variety  of  invariants  and  covariants  of  a  form  or  a 
set  of  forms.  We  shall  illustrate  it  by  means  of  several 
examples. 

Let  /  be  the  binary  quartic  and  let  cf>  be  the  form  f  itself. 
Then 

J         °dx*  1da%dx1  %dx$x\  *Bx2Bt\       *dx\ 

and 

■h  §f  '  /—  2(«0a4  —  4  aias  +  3  ai)  =  *'■ 

This  second  degree  invariant  i  represents  the  condition  that 
the  four  roots  of  the  quartic  form  a  self-ajDolar  range.  If 
this  process  is  applied  in  the  case  of  a  form  of  odd  order,  the 
result  vanishes  identically. 

If  ZTis  the  Hessian  of  the  quartic,  then 

d4  d4 

dH=  (a0a2  -  «f)—  -  2(a0a3  -  a^)- 


dx\        v  u  6        l  "dx\dx1 

Qi  ^4 

+  (a^  +  2  axaz  —  3  a|)  —  2(axa4  —  a2a3) 


di 

And 
^2  dJET  •/=  6(a0a2a4  +  2  a^ag  —  ajc?4  —  rt0a|  —  «|)  =  J".    (7(h) 

This  third-degree  invariant  equated  to  zero  gives  the  con- 
dition that  the  roots  of  the  quartic  form  a  harmonic  range. 
If  H  is  the  Hessian  of  the  binary  cubic /and 

then 

IdH-  g  =  [60(«ia3  -  rtD  +  hi(a\ch  ~  aoas)  +  k;( ' V'2  -  "?)>i 
+  [b^a^  -  a%)  +  b^a^  -  a0a3)  +  63(a0a2  -  a?)>2  ; 

a  linear  covariant  of  the  two  cubics. 


THE  PROCESSES   OF  INVARIANT   THEORY        51 


Bilinear  Invariants 

If    f=a0x'll+  •••is     a     binary     form     of     order 
g  =  b0x™  +  •••  another  of  the  same  order,  then 

1 


m    and 


df-g  =  a0bm  - 


|_    fl^m- 


+ 


+(-i)  :w 


in 


(71) 


+  ...  +(-l)ma„/v 

This,  the  bilinear  invariant  of  /  and  </,  is  the  simplest  joint 
invariant  of  the  two  forms.  If  it  is  equated  to  zero,  it  gives 
the  condition  that  the  two  forms  be  apolar.  If  m  =  2,  the 
apolarity  condition  is  the  same  as  the  condition  that  the  two 
quadratics  be  harmonic  conjugates  (Chap.  I,  §  1,  IV). 

VI.  The  fundamental   operation  called  trans vection.     The 

most  fundamental  process  of  binary  invariant  theory  is  a 
differential  operation  called  transvection.  In  fact  it  will 
subsequently  appear  that  all  invariants  and  covariants  of  a 
form  or  a  set  of  forms  can  be  derived  by  this  process.  We 
proceed  to  explain  the  nature  of  the  process.  We  first  prove 
that  the  following  operator  Q  is  an  invariant : 

^  _  dxx    dx2 
—(— 

where  (j/)  is  cogredient  to  (x).     In  fact  by  (70), 


(72) 


D,'  = 


x, 


dx. 


+  X, 


dx, 


o.r, 


()■>'; 


ty\      dv%     tyi      ty* 


=  (\H)CI, 


which  proves  the  statement. 

Evidently,  to  produce  any  result,  12  must  be  applied  to 
a  doubly  binary  function.  One  such  type  of  function  is  a 
y-polar  of  a  binary  form.     But 


52  THE   THEORY   OF   INVARIANTS 

Theorem.      The  result  of  operating  D,  upon  any  y-polar  of 
a  binary  form  f  is  zero. 
For,  if f=a0x™+  •  ••, 

m  —  r  \    dxj 


-1    dT+1f  (rX,r-l   dT+1f 


Hence 


—  rifi  x •••  —   -1  Wo 

dx[dx2  V1/   "     dx-^dx^ 

and  this  vanishes  by  cancellation. 

If  ft  is  operated  upon  another  type  of  doubly  binary  form, 
not  a  polar,  as  for  instance  upon/*?,  where/ is  a  binary  form 
in  xv  x2  and  g  a  binary  form  in  yv  yv  the  result  will  generally 
be  a  doubly  binary  invariant  formation,  not  zero. 

Definition.  If  f(x)  =  a0x™  -f-  •••is  a  binary  form  in  (x) 
of  order  m,  and  gQy)  =  b0y'{  +  •••  a  binary  form  in  (?/)  of  order 
n,  then  if  yv  y2  be  changed  to  xv  x2  respectively  in 

\m  —  r  \n  —  r 

=  nf(x)g(y),  (73) 

I  m  \n 

after  the  differentiations  have  been  performed,  the  result  is 
called  the  rth  trans vectant  (Cayley,  1816)  of  f(x)  and  g(x). 
This  will  be  abbreviated  (/,  g~)T,  following  a  well-established 
notation.     We  evidently  have  for  a  general  formula 

KJ'f  \m\n_       ^,  J\*jBa%"dal     dx\dx'2s 

We  give  at  present  only  a  few  illustrations.  We  note  that 
the  Jacobian  of  two  binary  forms  is  their  first  transvectant. 
Also  the  Hessian  of  a  form /is  its  second  transvectant.     For 


THE  PROCESSES   OF   INVARIANT   THEORY        53 
H= (  f     f      -  f  2    ^ 

9/"  1   NOV.'   Jt-Tli'    X->X«  J    J',1,/ 

m\m  —  iy 
fl»w  —  2 12 

_  K\"i        -)    ,f        f        -  2  f2      _i_  f       f       \ 

—  — ~, —^ V./  Xi.XiJ  x2zj  *V  .i'tij    i  .'  x2x2J  xxxx) 

As  an  example  of  multiple  transvection  we  may  write  the 
following  covariant  of  the  cubic/: 

Q  =  (/>(/>  /)2)1=  («0«3  ~  3  «0«1«2  +  2  ai>l 

(740 


+  3(«0a1a3  —  2  a0a|  +  a\a^)x\x2 


—  S(a0a2a3  —  2  «2a3  +  aja|  )x1a^ 

—  («0tf §  —  3  rt1«2«3  4-  2  a\)x\. 

If/  and  ^  are  two  forms  of  the  same  order  m,  then  (/,  g)m  is 
their  bilinear  invariant.  By  forming  multiple  transvections, 
as  was  done  to  obtain  Q,  we  can  evidently  obtain  an  un- 
limited number  of  concomitants  of  a  single  form  or  of  a  set. 

SECTION  2.     THE   ARONHOLD    SYMBOLISM.      SYMBOLICAL 
INVARIANT   PROCESSES 

I.    Symbolical  representation.     A  binary  form  /,   written 
in  the  notation  of  which 

/=  a^x\  +  3  axx\x2  +  3  a%%iX%  +  azx\ 

is  a  particular  case,  bears  a  close  formal  resemblance  to  a 
power  of  linear  form,  here  the  third  power.  This  resem- 
blance becomes  the  more  noteworthy  when  we  observe  that 

the  derivative  —  bears  the  same  formal  resemblance  to  the 
dx1 

derivative  of  the  third  power  of  a  linear  form  : 
df 


dxx 


%(aQx\  +  2  axxxx%  +  a2#§). 


That  is,  it  resembles  three  times  the  square  of  the  linear 
form.  When  we  study  the  question  of  how  far  this  formal 
resemblance  may  be  extended  we  are    led  to  a  completely 


54  THE   THEORY   OF   INVARIANTS 

new  and  strikingly  concise  formulation  of  the  fundamental 
processes  of  binary  invariant  theory.  Although/  =  a0x'{'  +  ••• 
is  not  an  exact  power,  we  assume  the  privilege  of  placing  it 
equal  to  the  with  power  of  a  purely  symbolical  linear  form 
a1x1  +  a2z2,  which  we  abbreviate  ax. 

This  may  be  done  provided  we  assume  that  the  only  defined 
combinations  of  the  symbols  uv  c^,  that  is,  the  only  combina- 
tions which  have  any  definite  meaning,  are  the  monomials 
of  degree  m  in  «r  «2; 


jin  —  l 
ll       "*2 


«0  =  a,.  ••-.  a"'  =  a, 


and.  linear  combinations  of  these.  Thus  a™  +  2  a™~2a%  means 
a0  +  2  av  But  af-2^  ^s  meaningless  :  an  umbral  expression 
(Sylvester).  An  expression  of  the  second  degree  like  a0as 
cannot  then  be  represented  in  terms  of  a's  alone,  since 
a™  •  a™_3«|  =  cqm_3a!  is  undefined.  To  avoid  this  difficulty  we 
give/* a  series  of  symbolical  representations, 

/=«™  =  #?  =  7»  =  ..., 

wherein  the  symbols  («r  «2),  (/3:,  /32),  (7l.  72),  •••  are  said  to 
be  equivalent  symbols  as  appertaining  to  the  same  form/. 
Then 

«?  =#r  -tt  =  •••  =^  «i"_1«2  =&?-102 =7i""172  =  •••  =<?r  •••• 

Now  aQas  becomes  (a'l'/3'{'~;i@f)  and  this  is  a  defined  combina- 
tion of  symbols. 

In  general  an  expression  of  degree  i  in  the  a's  will  be  repre- 
sented by  means  of  i  equivalent  symbol  sets,  the  symbols  of 
each  set  entering  the  symbolical  expressions  only  to  the  with 
degree ;  moreover  there  will  be  a  series  of  (equivalent) 
symbolical  representations  of  the  same  expression,  as 

a0as  =  af/3?-3/3§  =  <WtS  =  £WS7i  =  "" 


THE   PROCESSES   OF   INVARIANT   THEORY        55 

Thus  the  discriminant  of 

/=  <*!  =  £1  =  •••  =  a0xj  +  2  axxxx^  +  <z2z| 

D  =  4(a0a2  -  af)  =  4(«f/3§  -  a1a2@1@2') 
=  2(«?/3§-2«1<y31/32  +  «l/3f^ 

or  2>  =  2(«/3)2, 

a  very  concise  representation  of  this  invariant. 

Conversely,  if  we  wish  to  know  what  invariant  a  given 
symbolical  expression  represents,  we  proceed  thus.  Let/  be 
the  quadratic  above,  and 

g  =  pi  =  o-2  =   . ..  =  60zf  +  2  ^rr^  +  Vf, 

where  />  4s  not  equivalent  to  a.  Then  to  find  what 
J=  (ap^axPx,  which  evidently  contains  the  symbols  in  defined 
combinations  only,  represents  in  terms  of  the  actual  coeffi- 
cients of  the  forms,  we  multiply  out  and  find 

J=  (<*lP2  -  a2Pl)(alXl  +  a2X<i)(PlXl  +  P2X2) 

=  (alPlP2  -  ala2P\)X\  +  (alPl  —  alP'DXlX2  +  (ala2p2  -  a2plPz)XV 
=  ia0h\  ~  alh)Xl  +  (aoh  -  aMXlX2  +  (a\h2  -  a2^\)X2' 

This  is  the  Jacobian  of/ and  g.  Note  the  simple  symbolical 
form 

J=  {ap)axpx. 

II.  Symbolical  polars.  We  shall  now  investigate  the  forms 
which  the  standard  invariant  processes  take  when  expressed 
in  terms  of  the  above  symbolism  (Aronhold,  1858). 

For  polars  we  have,  when/=  a'"  =  yS™  =  •••, 

Hence 

fyr=a%-<-ary.  (75) 

The  transformed  form  of/ under  Twill  be 

/'  =  [«!( Vi'  +  A*i30  +  «2(  Vi  +  H^)T 

=  [(^Xj  +  «2X2)^  +  («x^i  +  «2/i'2)4]'"' 


«M 


56  THE   THEORY   OF  INVARIANTS 

or      /'  =  (>vi  +  V-2)W 

=  <.rl'"+  ...  +  r!)oif-rarllx!im-rx'2r+  •••  +a«4m.        (76) 

In  view  of  (75)  we  have  here  not  only  the  symbolical 
representation  of  the  transformed  form  but  a  very  concise 
proof  of  the  fact,  proved  elsewhere  (Chap.  I,  (29)),  that  the 
transformed  coefficients  are  polars  of  a'0=f  (Xr  X2)=  a™. 

The  formula  (66)  for  the  polar  of  a  product  becomes 

1  r 

,=        sX(m)(  n  yT'tfyPr+'ftr;     (7?> 

,'       fm  +  n\  7=<\sJ\r—sJ  J 

where  the  symbols  a,  /3  are  not  as  a  rule  equivalent. 

III.  Symbolical  transvectants.  If  /  =  «"'  =  ax—  •"■>  9  = 
fi%  =  bl=  .-.,  then  p  56 

/     32  d2     \ 

=  («/3)«r1^r1. 

Hence  the  symbolical  form  for  the  rth  transvectant  is 

(/,  gj  =  («/3)'-<-^rr-  (78) 

Several  properties  of  transvectants  follow  easily  from  this. 
Suppose  that  g  =  /  so  that  a  and  /3  are  equivalent  symbols. 
Then  obviously  we  can  interchange  a  and  ft  in  any  symbolical 
expression  without  changing  the  value  of  that  expression. 
Also  we  should  remember  that  (a/3)  is  a  determinant  of  the 
second  order,  wad.  formally 

(«/3)  =-(/?«). 

Suppose  now  that  r  is  odd,  r  =  2  k  +  1.     Then 

(ffYM  =  (a/S)2*^-2*-1/^"2*-1  =  -  (a^)2^1^-^-1^-2''-1. 

Hence  this  transvectant,  being  equal  to  its  own  negative, 
vanishes.     Every  odd  transvectant  of  a  form  with  itself  vanishes. 


THE   PROCESSES   OF  INVARIANT  THEORY        57 
If  the  symbols  are  not  equivalent,  evidently 

(/^)r=(-l)r(^/)r-  (79) 

Also  if  C  is  a  constant, 

(Cy,</)'=(7(/,<7)'-;  (80) 

Oi/i  +  °if-i  ^ — j  ^ii/i  +  ^2^2  +  •••)'' 

=  cA(fv  9i)'  +  Hd£fv  9i)"  +  ••-    (81> 

IV.  Standard  method  of  transvection.  We  may  derive 
transvectants  from  polars  by  a  simple  application  of  the 
fundamental  postulate.  For,  as  shown  in  section  1,  if/  = 
//   »•'"  4-  ...  —  //'" 

1  0    1      '  —      -r  ' 


/„r  = 


l£f>+{l)^te^  + 


drf     ' 


(82) 


Now  (y)  is  cogredient  to  (x).     Hence , are  cogredient 

dx%       dx^ 

to  yv  y2.     If  we  replace  the  y's  by  tliese  derivative  symbols 

and  operate  the  result,  which  we  abbreviate  as    df  r,  upon  a 

second  form  g  =  b"x,  we  obtain 


*fy 


aiK  -  ( i ) «rla2*r1Ji  +  •••  +(-1)r«26i  <~'^rr 
=  (a&y<-''&rr  =(f<ffy-  (83) 

When  we  compare  the  square  bracket  in  (82)  with  a'"-'- 
times  the  square  bracket  in  (83),  we  see  that  they  differ 
precisely  in  that  yv  y2  has  been  replaced  by  b2,  —  bv  Hence 
we  enunciate  the  following  standard  method  of  transvection. 
Let  /  be  any  symbolical  form.  It  may  be  simple  like  /  in 
this  paragraph,  or  more  complicated  like  (78),  or  howsoever 
complicated.  To  obtain  the  rth  transvectant  of  /and  Qf=  b^ 
we  polarize  f  r  times,  change  yv  y2  into  62,  —  b1  respectively  in 
the  result  and  multiply  by  bx~r.     In  view  of  the  formula  (77) 


58 


THE   THEORY   OF   INVARIANTS 


for  the  polar  of  a  product  this  is  the  most  desirable  method 
of  finding  transvectants. 

For  illustration,  let  F  be  a  quartic,  F  =  a%  =  bL  and  /  its 


Hessian 

» 

f=(abya%bl 

Let 

9  =  «!■ 

Then 

(f,gy  =  (abya2bl 

x  ax 

_(ab)2 
6 

'(o)©^ 

+  \lj{l)a*avh*bv  + 

©foK 

X  ttx  (84) 

=  ^(a&)2(6a)2a2ax-f-  |fa5)2(aa)(6a)a:z.62ax  +  ±(ab)2(aa)2b'].ux. 

Since  the  symbols  a,  b  are  equivalent,  this  may  be  simplified 
by  interchanging  a,  b  in  the  last  term,  which  is  then  identical 
with  the  first, 

(/,  <7)2=  K«5)2(*«)2«l«*  +  !(aJ)2(a«)(&a>A«*. 

By  the  fundamental  existence  theorem  this  is  a  joint  co- 
variant  of  F  and  g. 

Let/  be  as  above  and  g  =  («/3)«2/3x,  where  a  and  /3  are  not 
equivalent.  To  find  (/,  g~)2,  .say,  in  a  case  of  this  kind  we 
firStlGt  ^  =  («/3>lk=<r3, 

introducing  a  new  symbolism  for  the  cubic  g.  Then  we 
apply  the  method  just  given,  obtaining 

(/,  9Y  =  \(aby(b*yal*x  +  §( ab  )2(a<r)<>KVx- 

We  now  examine  this  result  term  by  term.  We  note  that 
the  first  term  could  have  been  obtained  by  polarizing  g  twice 
changing  yv  y2  into  bv  —  bx  and  multiplying  the  result  by 
(abyal 


Thus 
l(aby(b<rya**x  =  £(«/3)«5& 


(abya2..  (85) 


Consider  next  the  second  term.  It  could  have  been  obtained 
by  polarizing  g  once  with  regard  to  g,  and  then  the  result 
once  with  regard  to  z  ;  then  changing  gv  g2  into  a2,  —  av  and 


THE  PROCESSES   OF   INVARIANT   THEORY.       59 


zv  z2  into  bv  —  bv  and  multiplying  this  result  by  (ab)2axbx  ; 
|(a6)2(ao-)  (b<r}axbxax 


=  f  («£>!& 


From  (85), 


i  Ai  Wa+  UXor^3* 


jy=* 


z;  z=6 


( a5  )2a2 


X  (ab)2axbx.        (86) 


=  §(^)2(«/3)(«6)(/35)aX  +  i(«&)2(«/3X«&)2«3& 
From  (86), 


K«/3) 


oY1Wv+(i)(JWa 


s;  3=6 


(ab)2axb2 


=  |  af(£a)  +  i  «x/3x(«a) 


z;  z=b 


X  (a6)2(  «/3)aA 


=  2(«6)2(a/3)(a5)(/8a)«xaA  +  f(a6)2(a/3)(aa)  («&)&«  A 

+  %(aby(u/3)(/3b)(aa)axaxbx. 

Hence  we  have  in  this  case 

(/,  y)2  =  f (a&)2(a/3)(a&)(/3&)«2aa  +  l(a^)2(«/3)(^)2«2/8I 

+  |(a£)2(«/3) («&) (/8a) axaxbx  +  f (a&)2( «/3) (aa) («6)/3xa  A-  (87) 

V.   Formula  for  the  rth  transvectant.      The  most  general 
formulas  for/,  g  respectively  are  r 


in  which 


/=  <)a^  .J  4»\  g  =  /3n>/3<2)  ...  /3f , 


We  can  obtain  a  formula  of  complete  generality  for  the 
transvectant  (/,  g~)r  by  applying  the  operator  O  directly  to 
the  product//.     We  have 

d2        jj     _  >TA      (,j)  n(r)       fft 
Z.       7      J9  —  2-ia\    ^2        («)'/p(r)  ' 
3«l3y2  4   Py 


__52_ 

d:r23// 


,,     /? 


1  'V     A-\v 


60         -  THE   THEORY   OF   INVARIANTS 

Subtracting  these  we  obtain 

\m  —  1    n  —  1  __.  ftl 

Repetitions  of  this  process,  made  as  follows: 


im—  2  iw  —  2 


fg 


,  (88) 


lead  to  the  conclusion  that  the  rth  transvectant  of /a'nd  g,  as 
well  as  the  mere  result  of  applying  the  operator  £1  to  fg  r 
times,  is  a  sum  of  terms  each  one  of  which  contains  the 
product  of  r  determinant  factors  (a/3),  m  —  r  factors  «x,  and 
n  —  r  factors  fix.  We  can  however  write  (/,  g~)T  in  a  very 
simple  explicit  form.     Consider  the  special  case 

/=«<1>«<2>«<3),  g  =  &»&$>. 
Here,  by  the  rule  of  (88), 

(fg)2=  K«(1)/Sa))(«(2)/3'2))«'r3)  +  (a<i>£(1))(«f3)£(2))42) 
+  («(i'/3(2))(«(2,/3(1))«r3)  +  («(1,/3(20(«(3)/3(1))a<.2) 
+  (a^/^Xa^2')^  +  (a(2)/3(10(a(1)/3(2))433        (89) 
+  (a'2,/S(2))(«(3)/3,1))41)  +  («(2)^(2))(«'i)/3(i))u;;:» 
+  (a(3)^(D)(a(l)^2))^  +  («(3))g(l))(a(2)yQ(2))-aa) 

+  (««>/Sa))(«n)/8a))«a)  +  («(3,/3(2))(«(2)/3a))<1)$ -[2(3, 

in  which  occur  only  six  distinct  terms,  there  being  a  repetition 
of  each  term.  Now  consider  the  general  case,  and  the  rth 
transvectant.  In  the  first  transvectant  one  term  contains 
tl  =  («(1)/3(1,)42)  •••  4m)££2)  •••  /3£n).  In  the  second  transvectant 
there  will  be  a  term  ux=  («,1)/3(1,)(«(2)/3(2,)<43) ...  fif  ...  arising 
from  £ltv  and  another  term  ux  arising  from  fltv  where 
t2=(a^^)a^a'f  •••  a(Jn)/3M/3f  ...  £<»>.  Thus ux  (y  =  x)  and 
likewise  any  selected  term  occurs  just  twice  in  (/,#)2.     Again 

the  term  vx  =(a(i)yS(i))(«(2)^(2))("(s)^(S))44)  -  £*4)  —will 
occur  in  (/,  ^)3  as  many  times  as  there  are  ways  of  permuting 
the  three  superscripts  1,  2,  3  or  |3  times.     Finally  in  (/,  </)r, 


THE   PROCESSES   OF   INVARIANT   THEORY        61 

written  by  (88)  in  the  form  (89),  each  term  will  be  repeated 
[r  times.  We  may  therefore  write  (/,  g)r  as  the  following 
summation,  in  which  all  terms  are  distinct   and   equal    in 

)[r: 

•(«(1>^l>)(a^)/3(2))...ra(r)/8(,))  "I 

.  41)42)"-</3<1W).--/3(*    J9\y^ 


number  to 

r  J\r 


:Trk' 


VI.  Special  cases  of  operation  by  fl  upon  a  doubly  binary 
form,  not  a  product.  In  a  subsequent  chapter  Gordan's  series 
will  be  developed.  This  series  has  to  do  with  operation  by  fl 
upon  a  doubly  binary  form  which  is  neither  a  polar  nor  a 
simple  product.  In  this  paragraph  we  consider  a  few  very 
special  cases  of  such  a  doubly  binary  form  and  in  connection 
therewith  some  results  of  very  frequent  application. 

We  can  establish  the  following  formula: 

flr(.n/)r  =  constant  =  (r  +  l)(|r)2.  (91) 

In  proof  (74), 

fr-jc-iyf-)  *  ■    dr  -, 

and  (xyy  =  j>  (  -  V\Xrty[yV' 

i  =  0 

Hence  it  follows  immediately  that 

=  V([r)2=(r  +  l)(t)2. 

t'=0 

A  similar  doubly  binary  form  is 

F=(ryy&-ji;rj- 

If  the  second  factor  of  this  is  a  polar  of  ^+n'2J,  we  may 
make  use  of  the  fact,  proved  before,  that  12  on  a  polar  is  zero. 


62  THE   THEORY   OF   INVARIANTS 

An  easy  differentiation  gives 

£IF  =  j(m  +n-  j  +  V)(xy  y-^T%~\ 
and  repetitions  of  this  formula  give 

s  I  ?       \m  +  n  —  j  +  1         .....     ...         /If  i<  ;'• 

|y  — » |w  +  »  — .7  —  e  +  1  \  =  y)\ti>j 

(91.) 
This  formula  holds  true  if  m  =  w  =  /,  that  is,  for  fl'  (xy)j. 

VII.  Theorem.  Every  monomial  expression  (f>  which  con- 
sists entirely  of  symbolical  factors  of  two  types,  e.g.  determinants 
of  type  («/3)  and  linear  factors  of  the  type  ax,  and  which  is  a  de- 
fined expression  in  terms  of  the  coefficients  and  variables  of  a 
set  of  forms  f  g<  •••  is  a  concomitant  of  those  forms.  Con- 
versely, every  concomitant  of  the  set  is  a  linear  combination  of 
such  monomials. 

Examples  of  this  theorem  are  given  in  (78),  (84),  (87). 

In  proof  of  the  first  part,  let 

</,=(«/3K«7)'...«p^.., 

where  /=  a™;  and  /3,  7,  •••  may  or  may  not  be  equivalent  to 
a,  depending  upon  whether  or  not  cf>  appertains  to  a  single 
form/  or  to  a  set/,  g,  •••.  Transform  the  form/,,  that  is,  the 
set,  by  T.     The  transformed  of/  is  (76) 

f  =  (aKx\  +  aM4)m. 

Hence  on  account  of  the  equations  of  transformation, 

4>'  =  («a&  -  «m£a)"(<V)V  -  «m7a)9  •••  «£/%  .... 

But  «A/3M-«M/3A  =  (V)(«/3).  (92) 

Hence  <!>'  =  (Xfxy^-ff). 

which  proves  the  invariancy  of  (f>.  Of  course  if  all  factors 
of  the  second  type,  «T,  are  missing  in  cf>,  the  latter  is  an  in- 
variant. 

To  prove  the  converse  of  the  theorem  let  cf>  be  a  concomi- 


THE   PROCESSES   OF   INVARIANT   THEORY        63 

tant  of  the  set  /,  g,  •••  and  let  the  corresponding  invariant 
relation  be  written 

<£(<  a'v  •  ••  5  a£  4)  =  (X/A)^(a0,  ar  ... ;  a^,  £2).        (93) 

Now  aj  =  ttt~3'a^Q'  =  0,  1,  •••,  m).  Hence  if  we  substitute 
these  symbolical  forms  of  the  transformed  coefficients,  the 
left-hand  side  of  (93)  becomes  a  summation  of  the  type 

^PQx'^xlp  =  (\fi )k<f> (a0,  •••  ;  xv  x2)      (oj  +  o>2  =  &)),      (94) 

where  P  is  a  monomial  expression  consisting  of  factors  of  the 
type  «A  only  and  Q  a  monomial  whose  factors  are  of  the  one 
type  a^.  But  the  inverse  of  the  transformation  T  (cf.  (10)) 
can  be  written 


j.!  _      SV  /r.' SA 


CX/0  (V)' 

where  |2=  —  r2,  £2  =  -rr     Then  (94)  becomes 

xc-  i)^p^ci? = ( :xa*)*+-0.  o5) 

We  now  operate  on  both  sides  of  (95)  by  flfc+lu,  where 

n__* *_. 

We  apply  (90)  to  the  left-hand  side  of  the  result  and  (91) 
to  the  right-hand  side.  The  left-hand  side  accordingly  be- 
comes a  sum  of  terms  each  term  of  which  involves  neces- 
sarily &)  +  k  determinants  («/3),  («£).  In  fact,  since  the 
result  is  evidently  still  of  order  co  in  xv  xv  there  will  be  in 
each  term  precisely  <o  determinant  factors  of  type  («|)  and  k 
of  type  («/3).  There  will  be  no  factors  of  type  aK  or  £A  re- 
maining on  the  left  since  by  (91)  the  right-hand  side  becomes 
a  constant  times  <£,  and  (f>  does  not  involve  X,  /z.  We  now 
replace,  on  the  left,  («|)  by  its  equivalent  «x,  (/3£)  by  /3X,  etc. 
Then  (95)  gives,  after  division  by  the  constant  on  the  right^ 

<f>  =  la (a/3)p(  «7)«  •  •  •  «J/Sj  •  •  -,  (96) 

where  a  is  a  constant ;   which  was  to  be  proved. 


64  THE   THEORY   OF   INVARIANTS 

This  theorem  is  sometimes  called  the  fundamental  theorem 
of  the  symbolical  theory  since  by  it  any  binary  invariant 
problem  may  be  studied  under  the  Aronhold  symbolical 
representation. 


SECTIOX   3.     REDUCTIBILITY.     ELEMENTARY   COMPLETE 
IRREDUCIBLE   SYSTEMS 

Illustrations  of  the  fundamental  theorem  proved  at  the 
end  of  Section  2  will  now  be  given. 

I.  Illustrations.  It  will  be  recalled  that  in  (96)  each  sym- 
bolical letter  occurs  to  the  precise  degree  equal  to  the  order 
of  the  form  to  which  it  appertains.  Note  also  that  k  +  &>,  the 
index  plus  the  order  of  the  concomitant,  used  in  the  proof  of 
the  theorem,  equals  the  iveight  of  the  concomitant.  This 
equals  the  number  of  symbolical  determinant  factors  of  the 
type  (MJ3)  plus  the  number  of  linear  factors  of  the  type  ux  in 
any  term  of  cj>.  The  order  a>  of  the  concomitant  equals  the 
number  of  symbolical  factors  of  the  type  ax  in  any  term  of  <£. 
The  degree  of  the  concomitant  equals  the  number  of  distinct 
symbols  a,  /3.  •••  occurring  in  its  symbolical  representation. 

Let 

be  any  concomitant  formula  for  a  set  of  forms  /=  «"\ 
g  =  /3".  ••-.  No  generality  will  be  lost  in  the  present  dis- 
cussion by  assuming  0  to  be  monomial,  since  each  separate 
term  of  a  sum  of  such  monomials  is  a  concomitant.  In  order 
to  write  down  all  monomial  concomitants  of  the  set  of  a  given 
degree  i  we  have  only  to  construct  all  symbolical  products  (f> 
involving  precisely  i  symbols  which  fulfill  the  laws 

V  +  q  H +  p  =  m, 

(97) 
p  +  r  ^ -\-  a  —  n, 


THE   PROCESSES   OF   INVARIANT   THEORY 


65 


where,  as  stated  above,  m  is  the  order  of  /  and  equal  there- 
fore to  the  degree  to  which  a  occurs  in  <£,  w,  the  order  of  g, 
and  so  on. 

In  particular  let  the  set  consist  of  /=a2_/g2  merely. 
For  the  concomitant  of  degree  1  only  one  symbol  may  be 
used.  Hence  /=  a|  itself  is  the  only  concomitant  of  degree 
1.     If  i  =  2,  we  have,  for  0, 

and  from  (97) 

p  +  p  =  p  +  o-  =  2. 
Or 


p 

P 

cr 

0 

2 

2 

1 

1 

1 

2 

0 

0 

Thus  the  only  monomial  concomitants  of  degree  2  are 

<#M  =  A  («£)«*&  =  -  0/3X&  =  o,  («^)2  =  i  d- 

For  the  degree  3 

<f>  =  («/3)^(«7  )9(737)r«^7.;, 
jp  +  ^  +  ^  =  2,  j)  +  r  +  cr  =  2,   g-  +  r  +  t  =  2. 

It  is  found  that  all  solutions  of  these  three  linear  Diophan- 
tine  equations  lead  to  concomitants  expressible  in  the  form 
fsD\  or  to  identically  vanishing  concomitants. 

Definition.  Any  concomitant  of  a  set  of  forms  which  is 
expressible  as  a  rational  integral  function  of  other  concomi- 
tants of  equal  or  of  lower  degree  of  the  set  is  said  to  be 
reducible  in  terms  of  the  other  concomitants. 

It  will  be  seen  from  the  above  that  the  only  irreducible 
concomitants  of  a  binary  quadratic/  of  the  first  three  degrees 
are/ itself  and  Z),  its  discriminant.  It  will  be  proved  later 
that  /,  D  form  a  complete  irreducible  system  of  /.  By  this 
we  mean  a  system  of  concomitants  such  that  every  other  con- 
comitant of  /  is  reducible  in  terms  of  the  members  of  this 


66  THE   THEORY   OF  INVARIANTS 

system.  Note  that  this  system  for  the  quadratic  is  finite. 
In  another  chapter  we  shall  prove  the  celebrated  Grordan's 
theorem  that  a  complete  irreducible  system  of  concomitants 
exists  for  every  binary  form  or  set  of  forms  and  the  system 
consists  always  of  a  finite  number  of  concomitants.  All  of 
the  concomitants  of  the  quadratic  /  above  which  are  not 
monomial  are  reducible,  but  this  is  not  always  the  case  as  it 
will  be  sometimes  preferable  to  select  as  a  member  of  a  com- 
plete irreducible  system  a  concomitant  which  is  not  mono- 
mial (cf.  (87)).  As  a  further  illustration  let  the  set  of 
forms  be  /  =  a|  =  /3|  =  •••,  g  =  a2.  =  b2r  —  •  ••  ;  let  i  =  2. 
Then  employing  only  two  symbols  and  avoiding  («/3)2=|2), 
etc. 

p  +  P  =])  +  <r=  2. 
The  concomitants  from  this  formula  are, 

«2«2  —f .  g^   (aa)axax  =  J",   (art)2  =  h, 
J" being  the  Jacobian,  and  h  the  bilinear  invariant  of/ and  <£. 

II.  Reduction  by  identities.  As  will  appear  subsequently 
the  standard  method  of  obtaining  complete  irreducible  sys- 
tems is  by  transvection.  There  are  many  methods  of  prov- 
ing concomitants  reducible  more  powerful  than  the  one 
briefly  considered  above,  and  the  interchange  of  equivalent 
symbols.     One  method  is  reduction  by  symbolical  identities. 

Fundamental  identity.  One  of  the  identities  frequently 
employed  in  reduction  is  one  already  frequently  used  in 
various  connections,  viz.  formula  (92).     We  write  this 

axby  -  aybx  =  (ab)  (xy  ) .  (98) 

Every  reduction  formula  to  be  introduced  in  this  book,  in- 
cludirw  Grordan's  series  and  Stroh's  series,  may  be  derived 
directly  from  (98).  For  this  reason  this  formula  is  called 
the  fundamental  reduction  formula  of  binary  invariant  theory 
(cf.  Chap.  IV). 


THE   PROCESSES   OF   INVARIANT   THEORY        67 

If  we  change  yx  to  c2,  >/2  to  —  cv  (98)  becomes 

(5<?)aa.+(ca)Ja.+  (a6)ca.  =  0.  (99) 

Replacing  x  by  c?  in  (99), 

(ad)(bc)  4  (ca)(bd)  +  (ab)(cd)  =  0.  (100) 

From  (99)  by  squaring, 

2(a6)(ac)6xcx  =  (ai)2c2  +(ac)2*2  _(^)2a|.  (101) 

If  o)  is  an  imaginary  cube  root  of  unity,  and 

ux  =  (bc~)ax,  u2  =  (ca)bx,  u3  =  (ab)ex, 
we  have 

(wj  +  u2  +  w3)  (Wj  4-  &)W2  "+"  w2w3) (wi  +  &)2'/2  +  WM3 ) 

=  (ab'f4  +  (fo)3«|  +  (cayb%  —  3(a6)(6c)(m)az6IcI  =  0.     (102) 

Other  identities  may  be  similarly  obtained. 

In  order  to  show  how  such  identities  may  be  used  in  per- 
forming reductions,  let  f=a.x=b^.=  •••  be  the  binary  cubic 
form.     Then 

A  =  (/,/)2  =  («W*A, 

<?=(/,  A)  =  (ab)%cb)ax4. 

-  (/>  Q)2  =  \{ab)\bc)  \_axcl  4-  2  cxcyay~\ y=d  x  dx  (102.) 

=  %[{aby(edf(be)axdx  +  2(ai)2(a^)(e^)(6c)(?IcZ:r]. 

But  by  the  interchanges  a~d,  b  ~  c 

(ab)\cd)\be)axdx  =  (dc)20)*WM,=  0. 

By  the  interchange   c  ~  d  the    second  term    in  the    square 
bracket  equals 

(a£)2(^>vU<>0(/'<' )  4-  (ca)(W)], 

or,  by  (100)  this  equals 

{ab~)\cdycxdx  =  0. 

Hence  (/,  $)2  vanishes. 

We  may  note  here  the  result  of  the  transvection  (A,  A)2 ; 

i2=(A,  A)2=(a5)2(^)2(«0(^)- 


68 


THE    THEORY   OF   INVARIANTS 


III.  Concomitants  of  binary  cubic.  We  give  below  a  table 
of  transvectants  for  the  binary  cubic  form.  It  shows  which 
transvectants  are  reducible  in  terms  of  other  concomitants. 
It  will  be  inferred  from  the  table  that  the  complete  irredu- 
cible system  for  the  binary  cubic /consists  of 

/,  A,   £,  R, 

one  invariant  and  three  covariants,  and  this  is  the  case  as 
will  be  proved  later.  Not  all  of  the  reductions  indicated  in 
this  table  can  be  advantageously  made  by  the  methods  intro- 
duced up  to  the  present,  but  many  of  them  can.  All  four 
of  the  irreducible  concomitants  have  previously  been  derived 
in  this  book,  in  terms  of  the  actual  coefficients,  but  they  are 
given  here  for  convenient  reference : 

/=  aQx\  +  3  atxfx2  +  3  a2xxx\  +  azx\\ 

A  =  ~2(ct0a2  —  cif)x\  +  2(rt0^3  —  a^a^x^  +  2(aja3  —  a%)x% 

(cf.(35)), 

Q  =  (a%a3  —  3  aQaxa%  +  2  af)zf +  3(a0a1«3  —  2  a0a|  +  a^a^)x\x2 
—  3(a0a2tf3  —  2  a\az  +  a^a^x-p^  —  (#0a3  —  3  ata2a3  +  2  a%)x% 

(ci.  (39)), 

R  =8(«0a2  -  af)(a1as  -  a2,)-  2(a0a3  -  a^)2  (cf.  (74x)). 

The  symbolical  forms  are  all  given  in  the  preceding  Paragraph. 

TABLE   I 


First  Transv. 

Seconji  Tb  insv. 

Third  Tbansv. 

C/i/)  =  o 

(/./)"  =  A 

(/,/)*  =  o 

(/•  A)=  Q 

(/•  A)«  =  0 

(A.  A)=0 

(A,  A)2  =  22 

(/,  G)=-*a« 

(/,  #)'2  =  o 

(/,  qy  =  r 

(a.  Q)=  in/ 

(A,  Q)2  =  0 

(«,«)=  o 

i  Q,  Qy  =  J  BA 

(Q,Q)*  =  0 

THE    PROCESSES   OF   INVARIANT   THEORY        69 

SECTION  4.     CONCOMITANTS   IN    TERMS   OF   THE   ROOTS 

Every  binary  form  f  =  af  =  b™  =  •  ••  is  linearly  factorable 
in  some  field  of  rationality.     Suppose 

/=  (4%j  -  rFzJQrP*!  -  »f  )*2>"(>r)3a  -  r[m\~). 

Then  the  coefficients  of  the  form  are  the  elementary  sym- 
metric functions  of  the  m  groups  of  variables  (homogeneous) 

%/f>,  »#>)     0'  =  1,  2,  -..,  m). 

These  functions  are  given  by 

aj  =  (  -  iy  2  rprp  •  •  •  r[j)r%+1)  •  ■  •  r%n)     (J  =  0,  •  • .,  m) .   (103) 

The  number  of  terms  in  2  evidently  equals  the  number  of 
distinct  terms  obtainable  from  its  leading  term  by  permuting 
all  of  its  letters  after  the  superscripts  are  removed.  This 
number  is,  then, 

N=\m/\j  \m—j  =  mO}-. 

I.   Theorem.     Any  concomitant  of  f  is  a  simultaneous  con- 
comitant of  the  linear  factors  off  i.e.  of  the  linear  forms 

(/•<%),   (r(2)x),  •  ••,  (rim)x). 
For,  j.,  =  (_  i)»(r/(i)a/)(r/(2)a/)  ...  (r'(™V),  (104) 

and  a\  =  (-  lyZr'^r'™  ...  r'^r'^  •••  ^(m).       (1030 

Let  (f>  be  a  concomitant  of/,  and  let  the  corresponding  in- 
variant relation  be 

0'=(«f',,  •••,  a'my(x'v  x'2y = (\fiy\a0,  •••,oiOv^)u=(^)i^ 

When  the  primed  coefficients  in  cf>'  are  expressed  in  terms  of 
the  roots  from  (103j)  and  the  unprimed  coefficients  in  $  in 
this  invariant  relation  are  expressed  in  terms  of  the  roots 
from  (103),  it  is  evident  that  <f>'  is  the  same  function  of  the 
primed  r's  that  $  is  of  the  unprimed  r's.  This  proves  the 
theorem. 


70  THE   THEORY   OF   INVARIANTS 

II.  Conversion  operators.  In  this  Paragraph  much  advan- 
tage results  in  connection  with  formal  manipulations  by  in- 
troducing the  following  notation  for  the  factored  form  of/: 

f  =a(})a{})  •••  a'/'K  (105) 

Here   a(xj)  =  a[3')z1  +  aL^x2  (J  =  1,  •••,  ni).     The  as  are  related 

to  the   roots  (r{}),  r^y)   of    the  previous    Paragraph    by  the 

equations 

aU)  _  ry>j  ao)  _  _  r(i)  . 

that  is,  the  roots  are  (-Ha.y'),  —  a[jr)  (,/  =  l,  •••,»»).  The 
umbral  expressions  av  a2  are  now  cogredient  to  a[J\  «.y> 
(Chap.  I,  §  2,  VII,  and  Chap.  Ill,  (76)).     Hence, 

aU)  1_\  =  aU)  JL  +  «o»  J_ 
da/  dax        '    da2 

is  an  invariantive  operator  by  the  fundamental  postulate.  In 
the  same  way 

and  [Dabc...]  =  [Ba][I)b-][Dc]:. 

are  invariantive  operators.  If  we  recall  that  the  only  degree 
to  which  any  umbral  pair  av  a2  can  occur  in  a  symbolical 
concomitant, 

</)  =  2n&(a&)(ac)..., 

of  f  is  the  precise  degree  w,  it  is  evident  that  \_Dabc...~\  operated 
upon  <f>  gives  a  concomitant  which  is  expressed  entirely  in 
terms  of  the  roots  (a^,  —  a[j)~)  of  /.  Illustrations  follow. 
Let  2  <f>  be  the  discriminant  of  the  quadratic 

/=a|=51=...,^  =  (aJ)2. 

Then 

(a^—)<f>  =  2(a<i>&)(>&);   [Z)a]</>  =  2(a<i)&)(a<2)5). 

Hence 

[Dab]<f>=--2(uya'l)f.  (106) 


THE   PROCESSES    OF    INVARIANT   THEORY        71 

This  result  is  therefore  some  concomitant  of  f  expressed 
entirely  in  terms  of  the  roots  of/.  It  will  presently  appear 
that  it  is,  except  for  a  numerical  factor,  the  invariant  <£  it- 
self expressed  in  terms  of  the  roots.  Next  let  <\>  be  the  co- 
variant  Q  of  the  cubic /=  a§  =  •••.     Then 

Q  =  (aby(ac)bxcl, 
l[Z>a]  Q  =  (a^b)(a{i)b)Qca^bx4  +  (a^b)(a^b)(ca^)bx4 

KAd  Q  =  (aa)«(2))(«(2)aa))(ca(3))a(3)c2 

+  (  «(l)a(3,)(af2)«(1))(m(3>)«<2)6,2  +  (a(l)a(2))(a(2)a(8))(ca(8)  )aa>e2 
+  («(!>«  2))(aC3)ad))(ca(2)  ja(3)c2  +  (  a(Da(8))(a(8)a(l))(ca(2))a(2)^ 

+  («,i)«<3\)(a(3)«<20(c«(2V)<llc?  +  O<l3,«(1,)(rt,2,a(3))(^'1'  )42)<?| 

+  («<»««))  (a®  aa0(ca(1048)^+(«(8)a®)(a(2)a(80(ca(10^1)^ 

[Dabc]  #  =  _  252(a(1)a(2))2(aa)a(3))48)242)1  (107) 

wherein  the  summation  covers  the  permutations  of  the 
superscripts.  This  is  accordingly  a  covariant  of  the  cubic 
expressed  in  terms  of  the  roots. 

Now  it  appears  from  (104)  that  each  coefficient  of 
/  =  a%  =  •••  is  of  degree  m  in  the  «'s  of  the  roots  (a!/\  —  a[:,y). 
Hence  any  concomitant  of  degree  i  will  be  of  degree  im  in 
these  roots.  Conversely,  any  invariant  or  covariant  which  is 
of  degree  im  in  the  root  letters  a  will,  when  expressed  in 
terms  of  the  coefficients  of  the  form,  be  of  degree  i  in  these 
coefficients.  This  is  a  property  which  invariants  enjoy  in 
common  with  all  symmetric  functions.  Thus  [Z)„,,]c/>  above 
is  an  invariant  of  the  quadratic  of  degree  2  and  hence  it 
must  be  the  discriminant  cf>  itself,  since  the  latter  is  the 
only  invariant  of  /  of  that  degree  (cf.  §  3).  Likewise  it 
appears  from  Table  I  that  Q  is  the  only  covariant  of  the 
cubic  of  degree-order  (3,  3),  and  since  by  the  present  rule 
[I>abc~]Q  is  of  degree-order  (3,3),  (107)  is,  aside  from  a 
numerical  multiplier,  the  expression  for  Q  itself  in  terms  of 
the  roots. 


72  THE   THEORY   OF  INVARIANTS 

It  will  be  observed  generally  that  \_Dab...~\  preserves  not 
only  the  degree-order  (i,  w)  of  $,  but  also  the  weight  since 
10  =  i  (im  +  &)).  If  then  in  any  case  </>  happens  to  be  the 
only  concomitant  of/ of  that  given  degree-order  (i,  «o),  the 
expression  [2>o6...]</>  is  precisely  the  concomitant  <f>  expressed 
in  terms  of  the  roots.  This  rule  enables  us  to  derive  easily 
by  the  method  above  the  expressions  for  the  irreducible 
system  of  the  cubic  fin  terms  of  the  roots.     These  are 

/  =  a<%<2)af> ;   a*. 

A  =  S(«(i)a'2))2«(3)2  .    (abyaxbx. 

Q  =  2(a(i)«(2))2(«a)a(3))«<;,)2rt(2» .    (^)2(ac)JiC2. 

R  =  (a(1)«(2>)2(«(2^<3))2^(3;a(i))2  .    (aby(cd)\ac)(bd). 

Consider  now  the  explicit  form  of  Q : 

Q  =  («(l>a^)2(a(l)a(3))a©)2a(2)  +  (a(2)a(3))2(a(2)a(l))aa)2a«) 

+  («(3)«(i))2(«(3)«(2,)42)2<1)  +  («(3)«(2>)2(«^,c' ' '  ><Il2«f 

+  («(2)«<l))2^a(2)a(3))a(3)2aa)  +  («(l)a(3))2(a(l)a(2))a(2)2a(3). 

It  is  to  be  noted  that  this  is  symmetric  in  the  two  groups  of 
letters  (<*\j),  «2'))-  Also  each  root  (value  of/)  occurs  in  the 
same  number  of  factors  as  any  other  root  in  a  term  of  Q. 
Thus  in  the  first  term  the  superscript  (1)  occurs  in  three 
factors.     So  also  does  (2). 

III.  Principal  theorem.  We  now  proceed  to  prove  the 
principal  theorem  of  this  subject  (Cay ley). 

Definition.  In  Chapter  I,  Section  1,  II,  the  length  of 
the  segment  joining  0(xv  #2),  and  DQyv  y^)  ;  real  points, 
was  shown  to  be 

where  \  is  the  multiplier  appertaining  to  the  points  of 
reference  P,  Q,  and  /a  is  the  length  of  the  segment  PQ.  If 
the  ratios  xx :  xv  yx :  y2  are  not  real,  this  formula  will  not 
represent  a  real  segment  CD.  But  in  any  case  if  (r[3'\  r^), 
(r[k\  r2w)  are  any  two  roots  of  a  binary  form/=  a™,  real  or 


THE    PROCESSES   OF   INVARIANT   THEORY        73 

imaginary,  we  define  the  difference  of  these  two  roots  to  be 

the  number 

r>(i>r«)  i  =  XgO^V*') 

L  J      (Xrp  +  rpX^  +  ri*) 

We  note  for  immediate  use  that  the  expression 

II(r)  =  (Xr[v  +  r™  )( X/f>  +  ro))  ...  (\r«  +  r£»>) 

is  symmetric  in  the  roots.  That  is,  it  is  a  symmetric  func- 
tion of  the  two  groups  of  variables  (r[3\  r^})  (^  =  1,  •••,  m). 
In  fact  it  is  the  result  of  substituting  (1,  —X)  for  (xv  #2)  in 

/  =  (—  l)m(>(1).r)(r(2)^)  ...  (r(m):r), 
and  equals 

n(r)  =  (a0-wa1\+  •••  +(-l)mam\m). 

Obviously  the  reference  points  P,  Q  can  be  selected  *  so 
that  (1,  —  X)  is  not  a  root,  i.e.  so  that  II (r)  =f=  0. 

Theorem.     Let  f  be  any  binary  form,  then  any  function 
of  the    two  types   of  differences 

[r(i>r(*g,  [r^x]  =  \ii(r^x)/(\r^.  +  ^'))(X^1  +  z2), 

which  is  homogenous  in  both  types  of  differences  arid  symmetric 
in  the  roots  (r[j\  r2J'>)  (j  =  1,  •••,  m)  will,  when  expressed  in 
terms  of  xv  x2  and  the  coefficients  of  f  and  made  integral  by 
multiplying  by  a  poiver  of  II (r)  times  a  power  of  (X.Tj  +x2~), 
be  a  concomitant  if  and  only  if  every  one  of  the  products  of 
differences  of  which  it  consists  involves  all  roots  (r[j\  r^1 ' ) 
(values  of  j)  in  equal  numbers  of  its  factors.  Moreover  all 
concomitants  of  f  are  functions  (f>  defined  in  this  way.  If  only 
the  one  type  of  difference  [r0)  r(k)~\  occurs  in  <f>,  it  is  an  invari- 
ant, if  only  the  type  [_r(j)x],  it  is  an  identical  covariant, —  a 
power  of  f  itself,  and  if  both  types  occur,  <f>  is  a  covariant. 
[Cf.  theorem  in  Chap.  Ill,  §  2,  VII.] 

*  If  the  transformation  T  is  looked  upon  as  a  change  of  reference  points,  the 
multiplier  X  undergoes  a  homographic  transformation  under  T. 


74  tup:  theory  of  invariants 

Iii  proof  of  this  let  the  explicit  form  of  the  function  <b  de- 
scribed in  the  theorem  be 

<£=2t>:   '--VI'-1   '••  ]**— [r°  ajjft^a?]**—, 
where 

«i  +  0i  +  •••  =«2  +  y82+  —  =  •••• 

Pi  +  °"l  +    "'    =/32  +  °2  +    •"    =    *"• 

and  0  is  symmetric  in  the  roots.  We  are  to  prove  that  6  is 
invariantive  when  and  only  when  each  superscript  occurs  in 
the  same  number  of  factors  as  every  other  superscript  in 
a  term  of  <f>.  We  note  first  that  if  this  property  holds  and 
we  express  the  differences  in  </>  in  explicit  form  as  defined 
above,  the  terms  of  2  will,  without  further  algebraical  manip- 
ulation, have  a  common  denominator,  and  this  will  be  of  the 
form 

no-r,\,-1+,2 

Hence  H(r')u(\z1  +  r2y<f>  is  a  sum  of  monomials  each  one  of 
which  is  a  product  of  determinants  of  the  two  types  I  r  r  i, 
I  r  x).  But  owing  to  the  cogrediency  of  the  roots  and 
variables  these  determinants  are  separately  invariant  under 
T.  hence  Tl(r  )"(\-ri  +  ^y4>  i-s  a  concomitant.  Next  assume 
that  in  0  it  is  not  true  that  each  superscript  occurs  the  same 
number  of  times  in  a  term  as  every  other  superscript.  Then 
although  when  the  above  explicit  formulas  for  differences  are 
introduced  (\xx+x^)  occurs  to  the  same  power  v  in  every  de- 
nominator in  2.  this  is  not  true  of  a  factor  of  the  * 
(  \r[*  — /•.,  i.  Hence  the  terms  of  2  must  be  reduced  to  a 
common  denominator.  L^t  this  common  denominator  be 
n(r)w(^i  +  ^2)P-       Then   n  <,•,<  A,'!-  ./-.2)'6   is  of  the  form 

where  not  all  of  the  positive  integers  ui:.  are  zero. 


THE   PROCESSES    OF    INVARIANT    THEORY        75 

Nbw$j  is  invariantive  under  T.     Hence itmust  be  unaltered 

under  the  special   case  of    T:   x1  =  —  x^  x2  =  x[.      From   this 
r\ ■'  =  -  r.,  '.  /■,        /-,-  .     Hence 

0',  =  V  Yl  (  X^V'1  -  ^''  )V(>->' V^'  )X'-aV<:5r)^  •••  (  r^xyt  •.., 

*    j 
and  this  is  obviously  not  identical  with  (pl  on  account  of  the 
presence  of  the  factor  II.     Hence  ^>1  is  not  a  concomitant. 

All  parts  o(  the  theorem  have  now  been  proved  or  are  self- 
evident  except  that  <rfl  concomitants  of  a  form  are  expres- 
sible in  the  manner  stated  in  the  theorem.  To  prove  this. 
note  that  any  concomitant  <f>  of/,  being  rational  in  the  coeffi- 
cients of  /.  is  symmetric  in  the  roots.  To  prove  that  (/>  need 
involve  the  roots  in  the  form  of  differences  only,  before  it  is 
made  integral  by  multiplication  by  U.(r  ) "(  \.r{  +  .r.:  >»,  it  is 
only  necessary  to  observe  that  it  must  remain  unaltered 
when/'  is  transformed  by  the  following  transformation  oi 
determinant  unity  : 

•ri  =  x\  +  ''•''•:•  x-i  =  r'r 
and  functions  of  determinants  (r  >,  (r{  x)  are  the  only 

symmetric  functions  which  have  this  property. 

As  an  illustration  o(  the  theorem  consider  concomitants  of 

the  quadratic /=(V(1)#)(r(2)#).      'These  are  of  the  form 

0=2  l'"'V'"  J"'- 1  '"  '-''l"''  I  '■'■'■'']**• 
it 

Here  owing  to  homogeneity  in  the  two  types  o(  differences, 

rtl  =  (<-l  =    •"■  :     Pi  +  a\  ~  P-l  """  °8  —  '"• 

Also  due  to  the  fact   that   each  superscript    must  occur  as 
many  times  in  a  term  as  ever\  other  superscript, 

ul  -f  px  =  u,  -f  trv  <c,  -\-  p.y  =  <t.2  -f  <t.:.  .... 

Also   owing    to   s\  mmct  r\    u,   must    he  even.       Heme  «k=-«. 

p*  — «■*  =  A  and 

=  ,.;(r(P,.ej)>)2;,;^.llV)(  r<- V  )  , &  =  (7' 


76  THE   THEORY   OF   INVARIANTS 

where  O  is  a  numerical  multiplier.  Now  a  and  /3  may  have 
any  positive  integral  values  including  zero.  Hence  the 
concomitants  of /consist  of  the  discriminant  D=  —  (r(1V2))2, 
the  form/  =  (r(1)a;)(r(2)a;)  itself,  and  products  of  powers  of 
these  two  concomitants.  In  other  words  we  obtain  here  a 
proof  that  /  and  D  form  a  complete  irreducible  system  for 
the  quadratic.  We  may  easily  derive  the  irreducible  system 
of  the  cubic  by  the  same  method,  and  it  may  also  be  applied 
with  success  to  the  quartic  although  the  work  is  there 
quite  complicated.  We  shall  close  this  discussion  by  deter- 
mining by  this  method  all  of  the  invariants  of  a  binary  cubic 
/  =  -  (r  <%)  0(2)z)  (r™x) .      Here 


and 

That  is, 
Hence 


0  =  2  [r(1V(2)]aft[r(2)r(3)]p*[r(3V(1)]Y* 

k 

«*  +  7*  =  «*  +  &•  =  &•  +  7a- 
a*  =  &•  =  7a-  =  2  a. 


Thus  the  discriminant  R  and  its  powers  are  the  only 
invariants. 

IV.  Hermite"s  reciprocity  theorem.  If  a  form  f  —  a"!  =  b'"r 
=  •••  of  order  m  has  a  concomitant  of  degree  n  and  order  &>, 
then  a  form  g  =  a"  =  •••  °f  order  n  has  a  concomitant  of  degree 
m  and  order  w. 

To  prove  this  theorem  let  the  concomitant  of/ be 

1=  1k(aby(ac)Q  •••  arxb%  •••     (>  +  «+•••=  o>), 

where  the  summation  extends  over  all  terms  of  /  and  k  is 
numerical.  In  this  the  number  of  distinct  symbols  a,  b,  •••  is 
n.  This  expression  /  if  not  symmetric  in  the  n  letters 
a,b,c,  •••  can  be  changed  into  an  equivalent  expression  in  the 


THE   PROCESSES   OF   INVARIANT   THEORY        77 

sense  that  it  represents  the  same  concomitant  as  /,  and  which 
is  symmetric.     To  do  this,  take  a  term  of  Z,  as 

k(aby(acy  •••  arxb%  ••-, 

and  in  it  permute  the  equivalent  symbols  a,  &,  •••  in  all  \n 
possible  ways,  add  the  \n  resulting  monomial  expressions  and 
divide  the  sum  by  \n.  Do  this  for  all  terms  of  /  and  add 
the  results  for  all  terms.  This  latter  sum  is  an  expression  J 
equivalent  to  I  and  symmetric  in  the  n  symbols.  Moreover 
each  symbol  occurs  to  the  same  degree  in  every  term  of  J"  as 
does  every  other  symbol,  and  this  degree  is  precisely  m. 
Now  let 

#  =  41)42)  ...4»>, 

and  in  a  perfectly  arbitrary  manner  make  the  following  re- 
placements in  J : 

1    ,     b     ,     c     ,  ■••,  I 

c(i\      a^\      a(3\  ....  a(n\ 

The  result  is  an  expression  in  the  roots  («27\  —  aiJ))  °f  ffi 
Ja=  267(«(1)«(2))p(«(1,«(3,)5  •••  41,r42,s  •••, 

which  possesses  the  following  properties  :  It  is  symmetric 
in  the  roots,  and  of  order  &>.  In  every  term  each  root 
(value  of  O'))  occurs  in  the  same  number  of  factors  as 
every  other  root.  Hence  by  the  principal  theorem  of  this 
section  Ja  is  a  concomitant  of  g  expressed  in  terms  of  the 
roots.  It  is  of  degree  m  in  the  coefficients  of  g  since  it  is  of 
degree  m  in  each  root.     This  proves  the  theorem. 

As  an  illustration  of  this  theorem  we  may  note  that  a 
quartic  form  /  has  an  invariant  J  of  degree  3  (cf.  (70j))  ; 
and,  reciprocally,  a  cubic  form  g  has  an  invariant  R  of  degree 
4,  viz.  the  discriminant  of  g  (cf.  (39)). 


78  THE   THEORY   OF   INVARIANTS 

SECTION   5.      GEOMETRICAL    INTERPRETATIONS. 
INVOLUTION 

In  Chapter  I,  Section  1,  II,  III,  it  has  been  shown  how  the 
roots  (r['\  r|°)  (t'  =  l,  •••,  ni)  of  a  binary  form 

/=  («0,  av  •••,  amJxv  z2)m 

may  be  represented  by  a  range  of  m  points  referred  to  two 
fixed  points  of  reference,  on  a  straight  line  EF.  Now  the 
evanescence  of  any  invariant  of  /  implies,  in  view  of  the 
theory  of  invariants  in  terms  of  the  roots,  a  definite  relation 
between  the  points  of  this  range,  and  this  relation  is  such 
that  it  holds  true  likewise  for  the  range  obtained  from  /  =  0 
by  transforming/ by  T.  A  property  of  a  range  /=  0  which 
persists  for  f  =  0  is  called  a  projective  property.  Every 
property  represented  by  the  vanishing  of  an  invariant  I  of 
f  is  therefore  projective  in  view  of  the  invariant  equation 

I(a'0,  ...)=(\/x)A/(a0,  ...)• 

Any  covariant  of  f  equated  to  zero  gives  rise  to  a 
'*  derived  "  point  range  connected  in  a  definite  manner  with 
the  range /=  0,  and  this  connecting  relation  is  projective. 
The  identical  evanescence  of  any  covariant  implies  projec- 
tive relations  between  the  points  of  the  original  range  /=  0 
such  that  the  derived  point  range  obtained  by  equating  the 
covariant  to  zero  is  absolutely  indeterminate.  The  like 
remarks  apply  to  covariants  or  invariants  of  two  or  more 
forms,  and  the  point  systems  represented  thereby. 

I.    Involution.      If 

/=O0,  av  —Jxv  x2y\   f/  =  (b0.  bv  —Jzv  z2~)m 
are  two  binary  forms  of  the  same  order,  then 

/+  ^9  =  Oo  +  *fy><  (h  +  **r  ■•■lxv  %)m< 
where  k  is  a  variable  parameter,  denotes  a  system  of  qualities 
which  are  said  to  form,  with  f  and  g,  an   involution.      The 


THE   PROCESSES   OF  INVARIANT   THEORY        79 

single  infinity  of  point  ranges  given  by  k,  taken  with  the 
ranges  /  =  0,  g  =  0  are  said  to  form  an  involution  of  point 
ranges. 

In  Chapter  I,  Section  1,  V,  we  proved  that  a  point  pair 
((V),  (v))  can  be  found  harmonically  related  to  any  two  given 
point  pairs  ((p),  (>)),  ((?)>  (*))•  If  the  latter  two  pairs 
are  given  by  the  respective  quadratic  forms  /,  g,  the  pair 
((%),  (w))  is  furnished  by  the  Jacobian  Q  of  /,  g.  If  the 
eliminant  of  three  quadratics  /,  g,  h  vanishes  identically, 
then  there  exists  a  linear  relation 

f+kg  +  lh  =  0, 

and  the  pair  h  =  0  belongs  to  the  involution  defined  by  the 
two  given  pairs. 

Theorem.  There  are,  in  general,  2(?w  —  1)  quantics  he- 
longing  to  the  involution  f  +  kg  which  contain  a  squared  linear 
factor,  and  the  set  comprising  all  doable  roots  of  these  quantics 
is  the  set  of  roots  of  the  Jacobian  off  and  g. 

In  proof  of  this,  we  have  shown  in  Chapter  I  that  the  dis- 
criminant of  a  form  of  order  m  is  of  degree  2(ra  —  1). 
Hence  the  discriminant  of  f-\-  kg  is  a  polynomial  in  k  of 
order  2(m  —  1).  Equated  to  zero  it  determines  2(m  —  1) 
values  of  k  for  which/ -f  kg  has  a  double  root. 

We  have  thus  proved  that  an  involution  of  point  ranges 
contains  2(m  —  1)  ranges  each  of  which  has  a  double  point. 
We  can  now  show  that  the  2(m  —  1)  roots  of  the  Jacobian 
of/  and  g  are  the  double  points  of  the  involution.  For  if 
x^u2  —  x^i  is  a  double  factor  of  /  +  kg,  it  is  a  simple  factor 
of  the  two  forms 

V  +  k^  li  +  kX 


dx±  dx1      dx2         dx. 


and  hence  is  a  simple  factor  of  the  &- eliminant  of  these 
forms,  which  is  the  Jacobian  of  /,  g.  By  this,  for  instance, 
the  points  of  the  common  harmonic  pair  of  two  quadratics 


80  THE  THEORY   OF   INVARIANTS 

are  the  double  points  of  the  involution  denned  by  those 
quadratics.  The  square  of  each  linear  factor  of  C  belongs 
to  the  involution/  +  kg. 

In  case  the  Jacobian  vanishes  identically- the  range  of 
double  points  of  the  involution  becomes  indeterminate. 
This  is  to  be  expected  since  /  is  then  a  multiple  of  g  and  the 
two  fundamental  ranges  f =  0,  g  =  0  coincide. 

II.  Projective  properties  represented  by  vanishing  covari- 
ants.  The  most  elementary  irreducible  covariants  of  a  single 
binary  form  /=  (a0,  av  •••  Jxv  #2)m  are  the  Hessian  H,  and 
the  third-degree  covariant  T,  viz. 

H=  (/,  /)»   T=  (/,  H). 

We  now  give  a  geometrical  interpretation  of  each  of  these. 

Theorem.  A  necessary  and  sufficient  condition  in  order 
that  the  binary  form  f  may  be  the  mth  power  of  a  linear  form 
is  that  its  Hessian  H  should  vanish  identically. 

If  we  construct  the  Hessian  determinant  of  (r2x^  —  r^)™, 
it  is  found  to  vanish.  Conversely,  assume  that  K=  0. 
Since  IT  is  the  Jacobian  of  the  two  first  partial  derivatives 

-i—,   -^-,  the  equation  H=  0  implies  a  linear  relation 
dxx     ox2 


2dx, 

~Kl 

dx2 

=  0. 

Also 

by 

Euler's 

theorem 

df 

X-,  -=— 

1dx1 

+  x2 

dx2 

=  mf 

and 

df  7 
£^1  + 

dx2 

dx2  = 

=  df. 

Expansion  of  the  eliminant  of  these  three  equations  gives 

df =md(K^  +  K2*'2)^ 


THE   PROCESSES   OF   INVARIANT   THEORY        81 
and  by  integration 

and  this  proves  the  theorem. 

Theorem.  A  necessary  and  sufficient  condition  in  order 
that  a  binary  quartic  form  f=  a0r\  +  •••  should  be  the  product 
of  two  squared  linear  factors  is  that  its  sextic  covariant  T 
should  vanish  identically. 

To  give  a  proof  of  this  we  need  a  result  which  can  be  most 
easily  proved  by  the  methods  of  the  next  chapter  (cf. 
Appendix  (29))  e.g.  if  i  and  J  are  -the  respective  invariants 

of/, 

i  =  2(a0a4  —  4  axaz  +  3  a2,), 


J=6 
then 


A  1  wn 

a1     a2     a3 

tin  tt'o  t  It 


2       "3 

We  also  observe  that  the  discriminant  of  /  is  2t(z3—  6^2). 
Now  write  a2  as  the  square  of  a  linear  form,  and 

/=«&f.  =  4=6|=.... 


Then 


But 


Hence 


=  i[Oa)2?J  +  (?«)2«J  +  4(aa)(?aX2/|a! 
=  UKaaM  +  3(?a)*«?  -  2(«9)2a|]«|. 

(«a)%I  =  (/,  a|)2  =  l(«?)2a2, 

(?a)2«2  =  (/,  ^2)2  =  i[(«f/)2?2  +  3(^)2«|]. 


#  =  -  k»?)2/  +  K^)2»*-  (108> 

This  shows  that  when  .0"=  0,/  is  a  fourth  power  since  (aq)2, 
(qq~)2  are  constants. 

It  now  follows  immediately  that 

T=(fH)=l(qq)Xfax)a*. 


82  THE   THEORY   OF   INVARIANTS 

Next  if  /  contains  two  pairs  of  repeated  factors,  q2.  is  a 
perfect  square,  (qq*)2  =  0,  and  T=0.  Conversely,  without 
assumption  that  d\  is  the  square  of  a  linear  form,  if  ^=0, 
then 

and/  has  at  least  one  repeated  factor.  Let  this  be  ax.  Then 
from 

T=\(qq)Xf,ax)«l=Q, 

we  have  either  (<^)2  =  0,  when  q2.  is  also  a  perfect  square,  or 
(/,  ax)  =  0,  when/=  a4x.  Hence  the  condition  T=  0  is  both 
necessary  and  sufficient. 


CHAPTER   IV 

REDUCTION 
SECTION   1.     GORDAN'S   SERIES.     THE   QUARTIC 

The  process  of  making  reductions  by  means  of  identities, 
illustrated  in  Chapter  III,  Section  3,  is  tedious.  But  we 
may  by  a  generalizing  process,  prepare  such  identities  in 
such  a  way  that  the  prepared  identity  will  reduce  certain 
extensive  types  of  concomitants  with  great  facility.  Such 
a  prepared  identity  is  the  series  of  Gordan. 

I.  Gordan's  series.  This  is  derived  by  rational  operations 
upon  the  fundamental  identity 

«A  =  «A  +  O^X^y)* 
From  the  latter  we  have 

«*&;  =  OA  +  0A)O#)]"A-w     O^w) 

m     ,     s  (109) 

Since  the  left-hand  member  can  represent  any  doubly  binary 
form  of  degree-order  (m,  n'),  we  have  here  an  expansion  of 
such  a  function  as  a  power  series  in  (xy~).  We  proceed  to 
reduce  this  series  to  a  more  advantageous  form.  We  con- 
struct the  (n  —  k)th  y-polar  of 

(a»!,  h%y  ={ab  )ka';>-,:bnr-k, 

by  the  formula  for  the  polar  of  a  product  (66}.     This  gives 

_  _     (aby        "^  f    m  —  k    V    n  —  k     \       k_h  him-k-hin-m+h 
"  'm  +  n^2k\^i\m-k-h)\n-m  +  hJ  y  xx        °»         * 

n  —  k      J 

83 


84  THE   THEORY   OF   INVARIANTS 

Subtracting  Qab)kayl~':bxl~':bl-m  from  each  term  under  the 
summation  and  remembering  that  the  sum  of  the  numerical 
coefficients  in  the  polar  of  a  product  is  unity  we  immediately 
obtain 

~ C  "    x)y  (m  +  n-2 k\  & [m -k- h)\n - m  +  h, 

\      n  —  k      J 
x  a™-k-hb™-k-hl\-m  (ahrb>;,  -  «'fihx).  (ill) 

Aside  from  the  factor  f  ,  ]  the  left-hand  member  of  (111) 

is  the  coefficient  of  (xyy  in  (109).  Thus  this  coefficient  is 
the  (n  —  &)th  polar  of  the  kth  transvectant  of  a™ ,  J",  minus 
terms  which  contain  the  factor  (aby+l(xy~).  We  now  use 
(111)  as  a  recursion  formula,  taking  Jc=  m,  m  —  1,  •••.  This 
gives 

(aby-\bxbl-»>  =  (a>»,  b^pm+1-  -        —(a-  b%)™(xyy  (112) 

y  n  —  m  + 1  y 

We  now  proceed  to  prove  by  induction  that 

(aby+la™-k-lb>rk-lb^m  =  a0(  af,  b«  )*&_, 

+  «!«.  b»x)kZit_£xy)+  .» 

+«w-*-i«,  ^)^-?»(^/)m-fc-1, 

where  the  as  are  constants.     The  first  steps  of  the  induction 
are  given  by  (112).     Assuming  (113)  we  prove  that  the  rela- 
tion is  true  when  k  is  replaced  by  k  —  1. 
By  Taylor's  theorem 

p-l  +  £A-2+    ...    +|  +  1 

=  t^a-^y-l+th-2a-iy-2+  ...  +*1d-i)+*o- 

Hence 

(«#}  -  ajft*)  =  k_1(a&)*(sy)»+  ^2  W^)*- V*  +  •  " 

+  «A_t-(a6)A-i+1(^)A-i+14-1*r1+  -•  +f0(aJ)(^X1^-1. 

(114) 


REDUCTION  85 

Hence  (111)  may  be  written 

(ab')kanJ-kb™-kb,l-m=  (a™   bnrYyn_k 

m—k    h 

+  2  2  ^hi(^y~i+k+lKi~i~h+i~i^i~h+i~i^^(^y-i+\  (H5) 

in  which  the  coefficients  Ahi  are  numerical. 
But  the  terms 

Thi=(aby-i+k+la™-k-h+i-xb™-k-h+i-lb^m(r>i-k^  h  >  1,  e*  ^  A) 

for  all  values  of  A,  z  are  already  known  by  (112),  (113)  as 
linear  combinations  of  polars  of  transvectants ;  the  type  of 
expression  whose  proof  we  seek.  Hence  since  (115)  is  linear 
in  the  Thi  its  terms  can  immediately  be  arranged  in  a  form 
which  is  precisely  (113)  with  k  replaced  by  k  —  1.  This 
proves  the  statement. 

We  now  substitute  from  (113)  in  (109)  for  all  values  of  k. 
The  result  can  obviously  be  arranged  in  the  form 

«£&»  =  <70«\  bnxyyn  +  Ox(a™,  b%$r-i(xy)  +  ...  (116) 

+  (7,«,  b%yyn-j(xyy+  ...  +Om(af,  b-y-n_m{xyr. 

It  remains  to  determine  the  coefficients  (7,-.  By  (91  j)  of 
Chapter  III  we  have,  after  operating  upon  both  sides  of 
(116)  by  Q1  and  then  placing  y  =  x, 

I  m  I  n  \j  I  vi  -+-  n  —  j  +  1 

(abya'rjbrj  =  Gi  -         a . ,  v  w«ryflry- 


m—j    n—j 


m  +  n  —  2  j  +  1 


Solving  for   (?,•,  placing  the  result  in  (116)  (J  =  0,  1,  •••,  m), 
and  writing  the  result  as  a  summation, 

i=o  /w  +  n-.f  +  lN 

This  is  Gordan's  series. 

To  put  this  result  in  a  more  useful,  and  at  the  same  time 


86  THE   THEORY   OF   INVARIANTS 

a  more  general  form    let   us    multiply  (117)  by  (aby   and 
change  m.  n  into  m  —  r,  n  —r  respectively.     Thus 

(abya%-rb%-r 

fm  —  r\fn  —  r\ 

=  S/       \J         9  -Lt\  WO*<    Wy^-J-r-  ("8) 

~i  fm  +  n  —  Ir  —j  +  1\ 


j=H  _ 

If  we  operate  upon  this  equation  by  f  .r  —  J  ,  (y  —  J  ,  we  ob- 
tain the  respective  formulas 

(abyaf-rbkJb%-r-k 

fm  —  r\fn  —  r  —  k\ 

=  x  /  i  K  ?  .^l  o^yo*.  J*)ir-^-* ;     (H9) 

^  /  ?»  +  n  —  2  r  —  j  +  1  \  "   ' 

V  3 

(abya%-r-kaktb%-r 

fm  —  r  —  k\fn  —  r^ 

=  T  -^ '?         A    :? — L.  (xyVfcG,  6*  )^       _  (120) 


An  +  n  —  2r—j  +  1\ 


It  is  now  desirable  to  recall  the  standard  method  of  transvec- 
tion  ;  replace  yx  by  cT  y2  by  —  cx  in  (119)  and  multiply  by 
cv~n+r+k,  with  the  result 

(aby(bey-r-ka%-rbk<*-n+r+* 

fm  —  r\fn  —r  —  Jc 


=  T  (-  *)'/>     f      Ao      ^   •■L1N^a"'  ^);+'''  <£>n-''-r-*.        (121) 

\  3  ) 

Likewise  from  (120) 

(aby(bcy-r(acya™-r-kcv-n+r-k 
fm—  r  —  k\fn  —  r\ 

=  y  (-iy;     *  J^  *  I  (.(&%/"*  <€)•*-***•    (i22) 

"TT  fm+n  —  2r— y+i\ 

\  3  J 


SEDUCTION  87 

The  left-hand  member  of  equation  (121)  is  unaltered  in 
value  except  for  the  factor  (—  I)"-*  by  the  replacements 
a~c,  m~p,  r~n  —  r—k;  and  likewise  (122)  is  unaltered 
except  for  the  factor  (—  l)n+k  by  the  replacements  a~c, 
ra~p,  r~w  —  r.  The  right-hand  members  are  however 
altered  in  form  by  these  changes.  If  the  changes  are  made 
in  (121)  and  if  we  write  f  =  b%  g  =  a™,  h  =  eg,  ax  =  0,  a2  = 
n  —  r  —  Jc,  «o  =  r,  we  obtain 


HI 


ll 


m+  n  —  las—  j  +  1  \ 

5  J 

v~a\-  aNas\ 

=  (-i)a'T  7^ 4 —  •     1X  ((/» 7i)ai+y'  gT^3',    (123) 

^/w+jp-2«2-^  +  1\ 

where  we  have 

«2  +  «3  >  rc,  «3  +  ax  >  m,  «!  +  «2  ^J9'  (1240 

together  with  ax  =  0. 

If  the  corresponding  changes,  given  above,  are  made  in 
(122)  and  if  we  write  ax  =  k,  a2  =  n  —  r,  a3  =  r,  we  obtain  the 
equation  (123)  again,  precisely.  Also  relations  (121^)  repro- 
duce, but  there  is  the  additional  restriction  «2  +  «3  =  n. 
Thus  (123)  holds  true  in  two  categories  of  cases,  viz. 
(1)  «1  =  0  with  (1242),  and  (2)  «2  +  «3  =  n  with  (124j). 
We  write  series  (123)  under  the  abbreviation 

[f      9       A  ' 
n       m      p      ;   a2  +  «3>»1«3  +  «1>  m,  o^  +  a2 >p, 

«i       «2       a3 
(i)  «1  =  0, 
(ii)  «j  +  «2  =  w. 

It  is  of  very  great  value  as  an  instrument  in  performing 
reductions.  We  proceed  to  illustrate  this  fact  by  proving 
certain  transvectants  to  be  reducible. 


THE   THEORY   OF   INVARIANTS 


Consider  (A,  Q)  of  Table  I. 

(A,  <?)  =  ((A,/),A). 

Here  n  =  p  =  2,  m  =  3,  and  we  may  take  ax  =  0,  a2  =  «g  =  1, 
giving  the  series 

fA    /    A] 

2     3     2, 

,0     1     lj 
that  is, 

((A,/),  A)+f  ((A,/)2,  A)°=  ((A,  A),/)  +  K(A,  A)2,/)<>. 

But  (A,  A)  =  0,  (A,  /)2=  0,  (A,  A)2  =  R. 

Hence     (A,  Q)  =  ((A,  /),  A)  =  \  Rf, 

which  was  to  be  proved. 

Next  let/=a™  be  any  binary,  form  and  _0"=(/,/)2  its 
Hessian.  We  wish  to  show  that  ((/,  /)2,  /)2  is  always 
reducible  and  to  perform  the  reduction  by  Gordan's  series. 
Here  we  may  employ 

7  /  r 

m    m    m  , 

.0     3      1. 

and  since  (/,/)2*+1=  0,  this  gives  at  once 
m  -  1V3\  (m  -  1\(3\ 

2  m  —  2\  [2  to  -  4 \ 

i  ;  v  «  J 


Solving  we  obtain 
((/7)27)2  =  ^_ 


2w-6 
1 


^((/,/)V)°. 


((//)4,/)0 


m  -  3 


2(2  m  -  5) 


?r,    (124) 


2(2  m  —  o) 
where  i  =  (/,  /)4. 

Hence  when  ?n  >  4  this  transvectant  is  always  reducible. 


REDUCTION 


89 


II.  The  quartic.  By  means  of  Gordan's  series  all  of  the 
reductions  indicated  in  Table  I  and  the  corresponding  ones 
for  the  analogous  table  for  the  quartic,  Table  II  below,  can 
be  very  readily  made.  Many  reductions  for  forms  of  higher 
order  and  indeed  for  a  general  order  can  likewise  be  made 
(cf.  (124)).  It  has  been  shown  by  Stroh*  that  certain 
classes  of  transvectants  cannot  be  reduced  by  this  series  but 
the  simplest  members  of  such  a  class  occur  for  forms  of 
higher  order  than  the  fourth.  An  example  where  the 
series  will  fail,  due  to  Stroh,  is  in  connection  with  the 
decimic/=  ax°.     The  transvectant 

is  not  reducible  by  the  series  in  its  original  form  although 
it  is  a  reducible  covariant.  A  series  discovered  by  Stroh  will, 
theoretically,  make  all  reductions,  but  it  is  rather  difficult  to 
apply,  and  moreover  we  shall  presently  develop  powerful 
methods  of  reduction  which  largely  obviate  the  necessity  of 
its  use.  Stroll's  series  is  derived  by  operations  upon  the 
identity  (ab)cx  -f  (bc)ax  +  (ca)bx  =  0. 

TABLE    II 


/•  =  l 

,  =  2 

r  =  8 

r  =  4 

i 

(/,/)r 

0 

H 

0 

C/i  Hy 

T 

W 

0 

J 

(/,  Ty 

T^(ip-GH-) 

0 

\{Jf-iH) 

0 

(H,  HY 

0 

\(2Jf-iH) 

0 

K" 

(H,  TY 

i(jp-ifH) 

0 

k(W-6JH) 

0 

(2\  TY 

0 

—  7  2  ( <T2 + G  iH-— 12  JfH  ) 

0 

0 

We  infer  from  Table  II  that  the  complete  irreducible  sys- 
tem of  the  quartic  consists  of 

/,  H,  T,  i  J. 

*  Stroh  ;   Mathematische  Annalen,  vol.  31. 


90 


THE    THEORY   OF   INVARIANTS 


This  will  be  proved  later  in  this  chapter.  Some  of  this  set 
have  already  been  derived  in  terms  of  the  actual  coefficients 
(cf.  (70x)).  They  are  given  below.  These  are  readily 
derived  by  non-symbolical  transvection  (Chap.  Ill)  or  by 
the  method  of  expanding  their  symbolical  expressions  and 
then  expressing  the  symbols  in  terms  of  the  actual  coeffi- 
cients (Chap.  Ill,  §  2). 

/=  a0x\  +  4  a1x\x2  +  6  a2x%x\  +  4  azxxx\  +  aAx\, 
H=  2[(a0a2  -  a*)x\  +  2(a0a3  -  axa^)x\x2 
+  (a0a4  +  2  rtjtfg  —  3  a%)x\x^  +  2(«1a4  —  a2a3)  x-^x\  +  (a2a4  —  flg)^] , 

T= 
(afa3  —  3  a0ata2  +  2  af)x\  +  (a^«4  +  2  aQaxaz  —  9  a0a|  +  6  a\a2}x\x2 
+  5(a0a1a4  —  3  a0a2a3  +  2  a\a^)x\x\  +  10(afa4  —  a0af)x\xl 
+  5(  —  a0a3a4  +t3  axa2a±  —  2  a1a3r)x\x\  (125) 

+  (9  a4a|  —  a|a0  —  2  a^g^  —  6  a|a2)^1x| 
+  (3  a2a3a4  —  a^f  —  2  a|)a;|, 

»  =  2(a0a4  —  4  a1a3  +  3  a|), 


.7  =  6 


a0 

ax 

a2 

ai 

H 

H 

a2 

a3 

«4 

=  6(a0a2a4  +  2  a^ag  —  a|  —  aQa\  —  afa4) . 


These  concomitants  may  be  expressed  in  terms  of  the  roots 
by  the  methods  of  Chapter  III,  Section  4,  and  in  terms  of  the 
Aronhold  symbols  by  the  standard  method  of  transvection. 
To  give  a  fresh  illustration  of  the  latter  method  we  select 
T={f,H-)  =  -^HJ).     Then 

_  (aby 


=  I  (aby(bc)a*bx4+±(aby(ac)aM4 

=  (ab)\ac~)axb$c*. 


REDUCTION 


91 


Similar  processes  give  the  others.  We  tabulate  the  com- 
plete totality  of  such  results  below.  The  reader  will  find  it 
very  instructive  to  develop  these  results  in  detail. 

i  =  (aby, 
J=  (aby(be)2(ea)*. 

Except  for  numerical  factors  these  may  also  be  written 

iT=S(a<l)«(2))2a(3)2a(4)2? 


Cl)//(2)  \2/',/(l>/y(3)W2)/1,(3)2„(4)3 


<c\ 


"« 


(126) 


T=  2(a(i)a<2))2(a<1)a(s))a 

z  =  2(«(i)a(2))2(«<3,««))2, 
J=  2(a(1)a(2))2(a(3)a(4))2(a(3)a<1))(a(2)a(4) ). 

It  should  be  remarked  that  the  formula  (90)  for  the 
general  rth  trans vectant  of  Chapter  III,  Section  2  may  be 
employed  to  great  advantage  in  representing  concomitants 
in  terms  of  the  roots. 

With  reference  to  the  reductions  given  in  Table  II  we 
shall  again  derive  in  detail  only  such  as  are  typical,  to  show 
the  power  of  Gordan's  series  in  performing  reductions.  The 
reduction    of    (/,  Hy   has   been   given   above    (cf.  (124)). 

We  have 

(-  t,  Hy  =  ((^;/),  sy  =  (ir,  Ty. 

Here  we  employ  the  series 

H    f     H 
4      4      4. 
.0      3      1. 


This  gives 
'3\/3\ 


j=0        *   —.?\  j=0      * 


=Zfo-J 
9 


V) 


92  THE   THEORY   OF   INVARIANTS 

Substitution  of  the  values  of  the  transvectants  (.ff,/)r, 
(R,  Hy  gives 

(IT,  Tf  =  ^(-QJH+iJ). 

The  series  for  (21,  Tf=  ((/,  iT),  T7)2  is 

/  #  T7 

4  4  6    , 

0  2  1, 
or 

((/,  #),  20»  +  C(/,  #)2,  ?T)  =  ((/,  Z1)1.  #)  +  -!((/,  !F)W>. 
But 

((/,  5)2,   T)  =  (1  ft   7)  =  A  »(/.    ^)  =  Y2  (#"  -  6  ^2)- 

Hence,  making  use  of  the  third  line  in  Table  II, 

(T,  Ty  =  -  yV(*Y2  +  6  iW  -  12  JHf), 

which  we  wished  to  prove.  The  reader  will  find  it  profit- 
able to  perform  all  of  the  reductions  indicated  in  Table  II 
by  these  methods,  beginning  with  the  simple  cases  and  pro- 
ceeding to  the  more  complicated. 

SECTION   2.     THEOREMS   ON   TRANSVECTANTS 

We  shall  now  prove  a  series  of  very  far-reaching  theorems 
on  transvectants. 

I.  Theorem.  Every  monomial  expression,  <£>,  in  Aronhold 
symbolical  letters  of  the  type  peculiar  to  the  invariant  theory, 
i.e.  involving  the  two  types  of  factors  («£>),  ax; 

<f>  =  II(a&)pO?)«  •••  apJb°cTr  ..., 
is  a  term  of  a  determinate  transvectant. 

In  proof  let  us  select  some  definite  symbolical  letter  as  a  and 
in  all  determinant  factors  of  <j>  which  involve  a  set  a1  =  —  y2, 
a2  =  yv     Then  <f>  may  be  separated  into  three  factors,  i.e. 

4>'  =  PQ(& 


REDUCTION  93 

where  Q  is"  an  aggregate  of  factors  of  the  one  type  by, 
Q  =  bsyc*  •  ••,  and  P  is  a  symbolical  expression  of  the  same 
general  type  as  the  original  cf)  but  involving  one  less  sym- 
bolical letter, 

P  =  (bc)u(bdy  ->.bZel»: 

Now  PQ  does  not  involve  a.  It  is,  moreover,  a  term  of 
some  polar  whose  index  r  is  equal  to  the  order  of  <$  in  y. 
To  obtain  the  form  whose  rth  polar  contains  the  term  PQ  it 
is  only  necessary  to  let  y  =  x  in  PQ  since  the  latter  will 
then  go  back  into  the  original  polarized  form  (Chap.  Ill, 
§  1,  I).  Hence  (f>  is  a  term  of  the  result  of  polarizing 
(_PQ~)V=X  r  times,  changing  y  into  a  and  multiplying  this 
result  by  a%.  Hence  by  the  standard  method  of  transvec- 
tion  <f>  is  a  term  of  the  transvectant 

((PQ)y=x,  oT+ey     (r+p=m).  (127) 

For  illustration  consider 

cf,  =  (a5)2(ac)(ftc)axiJ.c|. 

Placing  a  ~  y  in  (a6)2(ac)  we  have 

<f>'  =-  b^ey(bo)bxcl .  ax. 

Placing  y~x  in<£'  we  obtain 

<£"  =  -  (be)b^ax. 

Thus  <f>  is  a  term  of 

In  fact  the  complete  transvectant  A  is 

+  A  =  -  ^(bc)(cayaM  -  i\(bc)(eay(ba~)axblcx 

-  io{l>c)(ca)(bayaxbxc%  -  ^\(bc~)(ba)saxc^ 

and  </>  is  its  third  term. 

Definition,  The  mechanical  rule  by  which  one  obtains 
the  frgiwfovootnrwt  (a6)a"i_16".t_1  from  the  product  a£b%,  consist- 
ing of  folding  one  letter  from  each  symbolical  form  a™,  b™ 


94  THE   THEORY   OF   INVARIANTS 

into  a  determinant  (a5)  and  diminishing  exponents  by  unity, 
is  called  convolution.  Thus  one  may  obtain  (a6)2(ac)aa.5|c| 
from  («6)a|J|c|  by  convolution. 

II.   Theorem.     (1)    The  difference  between  any  two  terms  of  a 
transvectant  is  equal  to  a  sum  of  terms  each,  of  which  is  a  termm 
of  a  transvectant  of  lower  index  of  forms  obtained  from   the 
forms  in  the  original  transvectant  by  convolution. 

I  2  )  The  difference  between  the  whole  transvectant  and  one 
of  its  terms  is  equal  to  a  sum  of  terms  each  of  which  is  a  term 
of  a  transvectant  of  lower  index  of  forms  obtained  from  the 
original  forms  by  convolution  (Gordan). 

In  proof  of  this  theorem  we  consider  the  process  of  con- 
structing the  formula  for  the  general  rth  transvectant  in 
Chapter  III,  Section  5.  In  particular  we  examine  the 
structure  of  a  transvectant-like  formula  (89).  Two  terms 
of  this  or  of  any  transvectant  are  said  to  be  adjacent  when 
they  differ  only  in  the  arrangement  of  the  letters  in  a  pair 
of  symbolical  factors.  An  examination  of  a  formula  such  as 
(89)  shows  that  two  terms  can  be  adjacent  in  any  one  of 
three  ways,  viz. : 

(1)  P(«"',/3(','))(«(A)/3a',)  and  P(«(')/3(W)(a(W^'>), 

(2)  P(aP>/3&)aW        and  P(«(/')y8(^)<'. 

(3)  Piy0/^)/^'1        and  P(a®frk))pj\ 

where  P  involves  symbols  from  both  forms  /,  g  as  a  rule, 
and  both  types  of  symbolical  factors. 

The  differences  between  the  adjacent  terms  are  in  these 
cases  respectively 

(l).P(«(',<t"'))(/3(',/3<x-)), 

(2)  P(«(<V/',)/31/'. 

(3)  POS^/^'K". 

These  follow  directly  from  the  reduction  identities,  i.e.  from 
formulas  (99),  (100). 


REDUCTION  05 

Now,  taking  /,  g  to  be  slightly  more  comprehensive  than 
m  (89),  let 

f=Aa<pa®>  ...<J), 

g  =  B/3™/3™  •••  /3y", 

where  A  and  B  involve  only  factors  of  the  first  type  (7S). 
Then  formula  (90)  holds  true  ; 


/  m  \  /  V),  \  ^^ 


m\  n 
r 
1 \  r  Ar 


'(«(1)/Q(1))(«(2)/3(2))  •••  (a<r)0ir))  f  ' 


and  the  difference  between  any  two  adjacent  terms  of  (/,  #)r 
is  a  term  in  which  at  least  one  factor  of  type  (aft)  is  re- 
placed by  one  of  type  (««')  or  of  type  (/3/3').  There  then 
remain  in  the  term  only  r—1  factors  of  type  (a/3).  Hence 
this  difference  is  a  term  of  a  transvectant  of  lower  index  of 
forms  obtained  from  the  original  forms/",  g  by  convolution. 
For  illustration,  two  adjacent  terms  of  ((a&)2a|6|,  cj.)2  are 

(ab)\acyblcl  (aby(ac-)(bc)axbxc*. 

The  difference  between  these  terms,  viz.  (ab)s(ac)bxc%  is  a 
term  of 

aabfaj^  C|), 

and  the  first  form  of  this  latter  transvectant  may  be  obtained 
from  (ab)2a2.b2.  by  convolution. 

Now  let  tv  t2  be  any  two  terms  of  (/,  g)T.  Then  we  may 
place  between  tv  t2  a  series  of  terms  of  (/,  <jr)r  such  that  any 
term  of  the  series, 

^l'  hv  ^12'  '"  hi'  h 

is  adjacent  to  those  on  either  side  of  it.  For  it  is  always 
possible  to  obtain  t2  from  tx  by  a  finite  number  of  inter- 
changes of  pairs  of  letters,  —  a  pair  being  composed  either 
of  two  «'s  or  else  of  two  /3's.     But 

h-h  =  (h  -  hi)  +  c/n  -  «a)  +  -  +  (hi  -  *2)> 


96  THE   THEORY   OF   INVARIANTS 

and  all  differences  on  the  right  are  differences  between 
adjacent  terms,  for  which  the  theorem  was  proved  above. 
Thus  the  part  (1)  of  the  theorem  is  proved  for  all  types  of 
terms. 

Next  if  t  is  any  term  of  (/,  y)T,  we  have,  since  the  number 
of  terms  of  this  transvectant  is 

m\fn 
r  J\r 

(fyY-t  = - Zt'-t  (128) 


Ir 


m\(n 
r  )\r 


1 


r 


m\/n 
r  j\r 


2(f  -  0. 


and  by  the  first  part  of  the  theorem  and  on  account  of  the  form 
of  the  right-hand  member  of  the  last  formula  this  is  equal  to 
a  linear  expression  of  terms  of  transvectants  of  lower  index 
of  forms  obtained  from/,  y  by  convolution. 

III.  Theorem.  The  difference  between  any  transvectant  and 
one  of  its  terms  is  a  linear  combination  of  transvectants  of 
lower  index  of  forms  obtained  from  the  oriyinal  forms  by 
convolution. 

Formula  (128)  shows  that  any  term  equals  the  transvec- 
tant of  which  it  is  a  term  plus  terms  of  transvectants  of 
lower  index.  Take  one  of  the  latter  terms  and  apply  the 
same  result  (128)  to  it.  It  equals  the  transvectant  of 
index  s<r  of  which  it  is  a  term  plus  terms  of  transvectant 
of  index  <  s  of  forms  obtained  from  the  original  forms  by 
convolution.  Repeating  these  steps  we  arrive  at  transvec- 
tants of  index  0  between  forms  derived  from  the  original 
forms  by  convolution,  and  so  after  not  more  than  r  applica- 
tions of  this  process  the  right-hand  side  of  (128)  is  reduced 
to  the  type  of  expression  described  in  the  theorem. 

Now  on  account  of  the  Theorem  I  of  this  section  we  may 


REDUCTION  97 

go  one  step  farther.  As  proved  there  every  monomial  sym- 
bolical expression  is  a  term  of  a  determinate  transvectant 
one  of  whose  forms  is  the  simple  /  =  af  of  degree-order 
(1,  m).  Since  the  only  convolution  applicable  to  the  form 
a™  is  the  vacuous  convolution  producing  a™  itself,  Theorem 
III  gives  the  following  result  : 

Let  <f)  be  any  monomial  expression  in  the  symbols  of  a 
single  form  /,  and  let  some  symbol  a  occur  in  precisely  r 
determinant  factors.  Then  (f>  equals  a  linear  combination 
of  trans vectants  of  ind&x-<  r  of  a™  and  forms  obtained  from 
(PQ)y=x  (cf.  (127))  by  convolution. 

For  illustration 

<j>=(aby(bcya$c*=((abya2bl  4)2 -((«&)%  A,  4) 

+  1((^)4,  4)°- 
It  may  also  be  noted  that  (PQ)y=x  and  all  forms  obtained 
from  it  by  convolution  are  of  degree  one  less  than  the  degree 
of  <f>  in  the  coefficients  of  /.  Hence  by  reasoning  induc- 
tively from  the  degrees  1,  2  to  the  degree  i  we  have  the 
result  : 

Theorem.  Every  concomitant  of  degree  i  of  a  form  f  is 
given  by  transvectants  of  the  type 

where  the  forms  C^  are^'ail  concomitants  of  f  of  degree  i  —  1. 
(See  Chap.  Ill,  §  2,  VII.) 

SECTION   3.     REDUCTION   OE   TRANSVECTANT   SYSTEMS 
We  proceed  to  apply  some  of  these  theorems. 

I.  Reducible  transvectants  ( C7^!, /)y.  The  theorem  given 
in  the  last  paragraph  of  Section  2  will  now  be  amplified  by 
a*©the*  proof.  Suppose  that  the  complete  set  of  irreducible 
concomitants  of  degrees  <  i  of  a  single  form  is  known.  Let 
these  be 

/>  7r  72>    *•'  •/*> 


98  THE    THEORY    OF    INVARIANTS 

and  let  it  be  required* to  rind  all  irreducible  concomitants  of 
degree  i.  The  only  concomitant  of  degree  unity  is/=a™. 
All  of  degree  2  are  given  by 

(f,fy=(abya>rTb™-r, 

where,  of  course,  r  is  even.  A  covariant  of  degree  i  is  an 
aggregate  of  symbolical  products  each  containing  i  symbols. 
Let  Ct  be  one  of  these  products,  and  a  one  of  the  symbols. 
Then  by  Section  2  Ct  is  a  term  of  a  transvectant 

where  Cf_i  is  a  symbolical  monomial  containing  i  —  1  sym- 
bols, i.e.  of  degree  i  —  1.     Hence  by  Theorem  II  of  Section  2, 

where  (7$_i  is  a  monomial  derived  from  Ci_\  by  convolution. 
Now  (/$_!,  CVi  being  of  degree  i  —  1  are  rational  integral 
expressions  in  the  irreducible  forms  /,  yv  •  ••,  yk.  That  is, 
they  are  polynomials  in  /,  yv  ••■,  7*.,  the  terms  of  which  are 
of  the  type 

<k-i=/°7?  •••  7**- 

Hence  Ct  is  a  sum  of  transvectants  of  the  type 

Ohr-vfy  c/<»), 

and  since  any  covariant  of/,  of  degree  i  is  a  linear  combina- 
tion of  terms  of  the  type  of  C&  all  concomitants  of  degree  i 
are  expressible  in  terms  of  transvectants  of  the  type 

(fc-n/X  (130) 

where  </>;_!  is  a  monomial  expression  in  /,  7r  •••,  yk,  of  degree 
i  —  1,  as  just  explained. 

In  order  to  find  all  irreducible  concomitants  of  a  stated 
degree  i  we  need  now  to  develop  a  method  of  finding  what 
transvectants  of  (130)  are  reducible  in  terms  of  /,  7r  •••,  7*.. 
With  this  end  in  view  let  <f>i_1  =  po-*  where  p,  a  are  also 
monomials  in/,  yv  •••,  7^,  of  degrees  <  1— 1.     Let   p   be    a 


REDUCTION  99 

form  of  order  nx ;  p  =  p'*,  and  a  =  anx\  Then  assume  that 
j  —  n2->  ^ie  order  of  a.     Hence  we  have 

Then  in  the  ordinary  way  by  the  standard  method  of  trans- 
vection  we  have  the  following : 

=  KP(<T,fy+.~.  cm) 

Hence   if  p2  now  represents  (<r,  /)•%  then  pp2  is  a  term  of 

(&_i,/y  =^2  +  MA>i-x,fY    (f </).        (132) 

Evidently  p,  /a2  are  both  covariants  of  degree  <  i  and  hence 
are  reducible  in  terms  of  /,  yv  •••,  yk.  Now  we  have  the 
right  to  assume  that  we  are  constructing  the  irreducible  con- 
comitants of  degree  i  by  proceeding  from  transvectants  of  a 
stated  index  to  those  of  the  next  higher  index,  i.e.  we 
assume  these  transvectants  to  be  ordered  according  to  in- 
creasing indices.  This  being  true,  all  of  the  transvectants 
(0i-u  /);  at  the  stage  of  the  investigation  indicated  by 
(132)  will  be  known  in  terms  of  /,  yv  •••,  y*  or  known  to  be 
irreducible,  those  that  are  so,  since  j'  <j-  Hence  (132) 
shows  (<pi-i,f)  to  be  reducible  since  it  is  a  polynomial  in 
/,  yv  •••,  yk  and  such  concomitants  of  degree  i  as  are  already 
known. 

The  principal  conclusion  from  this  discussion  therefore  is 
that  irreducible  concomitants  of  degree  i  are  obtained  only 
from  transvectants  (<£i_1,/y  for  which  no  factor  of  order  ^j 
occurs  in  <^>f_1.  Thus  for  instance  if  m  =  4,  (/2,/)J  is  re- 
ducible for  all  values  of  j  since/2  contains  the  factor/  of 
order. 4  and/  cannot  exceed  4. 

We  note  that  if  a  form  y  is  an  invariant  it  may  be  omitted 
when  we  form  <£,-_!,  for  if  it  is  present  (0,-_i,/y  will  be  re- 
ducible by  (80). 


100 


THE   THEORY   OF   INVARIANTS 


II.  Fundamental  systems  of  cubic  and  quartic.  Let  m=3 
(cf.  Table  I).  Then  /=  a|  is  the  only  concomitant  of 
degree  1.  There  is  one  of  degree  2,  the  Hessian  (/,/)2  =  A. 
Now  all  forms  <£2  of  (<£2./y  are  included  in 

c/>2=/*AS, 

and  either  «  =  2,  fi  =  0,  or  a  =  0,  £=  1.  But  (/2,/);  is  re- 
ducible for  all  values  of  y  since/2  contains  the  factor  /  of 
order  3  and  j  >  3.  Hence  the  only  transvectants  which 
could  give  irreducible  concomitants  of  degree  3  are 

(A,/y   o'  =  i,  2). 

But  (A,/)2  =  0  (cf.  Table  I).     In  fact  the  series 

[/    /    /] 
3     3     3 
12     1 

gives       K(//)2'/)2  =  -(a/)2^/)2=-(A,/)2  =  0. 

Hence  there  is  one  irreducible  covariant  of  degree  3,  e.g. 

(Xf)  =  -Q. 

Proceeding  to  the  degree  -I,  there  are  three  possibilities 
for  <f>3  in  (<f>yfy.  These  are  <£3  =/3,  /A,  Q.  Since  />3 
(/W<  (/A,/y  (i=l,  2,  3)  are  all  reducible  by  Section  3, 1. 
Of  (<>,  /y  Q'  =  1,  2,  3),  (Q,  /)2  =  0,  as  has  been  proved 
before  (cf.  (102)),  and  ($,/)  =  iA2  by  the  Gordan  series 
(cf.  Table  I) 

[/    A    /I 
3     2     3 

v°     !     *, 

Hence  (Q,f)3  =  —  B  is  the  only  irreducible  case.  Next  the 
degree  5  must  be  treated.     We  may  have 

<k=/*/2A,/&i2,A2. 

But  R  is  an  invariant,  A  is  of  order  2,  and  Q  of  order  3. 
Hence   since  y>  3   in   (<£4,  f)j   the   only  possibility  for   an 


101 


REDUCTION 

irreducible  form  is  (A2,  /y,  and  this  is  reducible  by  the  prin- 
ciple of  I  if  j  <  3.     But  y  Ufc^  £*.  K*}**-  ' " 

( A2,  f  )«  =  (S|8£  asy=  (8ay(B'a)8'x  =  (8£  (Sa)2ax)  =  0. 

For  (S«)2ax  =  (A,  /)2  =  0,  as  shown  above.  Hence  there  are 
no  irreducible  concomitants  of  degree  5.  It  immediately 
follows  that  there  are  none  of  degree  >  5,  either,  since  <f>6  in 
($&fy  is  a  niore  complicated  monomial  than  04  in  the  same 
forms/,  A,  Q  and  all  the  resulting  concomitants  have  been 
proved  reducible. 

Consequently  the  complete  irreducible  system  of  concom- 
itants of  /,  which  may  be  called  the  fundamental  system 
(Salmon)  of /is 

/,  A,   Q,  E. 

Next  let  us  derive  the  system  for  the  quartic  / ;  m  =  4. 
The  concomitants  of  degree  2  are  (/,/)2=^,  (//)4  =  i. 
Those  of  degree  3  are  to  be  found  from 

{Hjy     C/  =  l,  2,  3,4). 

Of  these  (/,  H)  =  T,  and  is  irreducible  ;  (/,  Hy  =  J  is  irre- 
ducible, and,  as  has  been  proved,  (H,fy  =  ^if  (cf.  (124)). 
Also  from  the  series 


/ 

/ 

/' 

4 

4 

4 

1 

o 
O 

1 

(i/,/)3  =  0.     For  the  degree  four  we  have  in  ($3,  /)> 

03  =/3,  fff,   % 

all  of  which  contain  factors  of  order  ~>j  >  4  except  T. 
From  Table  II  all  of  the  transvectants  (27,/)1'  0'  =  1,  2, 
3,  4)  are  reducible  or  vanish,  as  has  been,  or  may  be  proved 
by  Gordan's  series.  Consider  one  case ;  (T7,/)4.  Applying 
the  series 

f    H  / 

4     4     4 

13     1 


L 


'102  the  theory  of  invariants 

we  obtain 

((/,  H),  /y  =  -((/,  H)\fy  -  ^((/,  j?)3,/)2. 

But  ((/,  H)\ff=\i(f,fy=i);  and  (/,  H)*  =  0  from 
the  proof  above.     Hence 

((/,#),/)*  =  (Ti/)4  =  0. 

There  are  no  other  irreducible  forms  since  </>4  in  (<£4, /)y  will 
be  a  monomial  in/,  5",  7  more  complicated  than  </>3.  Hence 
the  fundamental  system  of/  consists  of 

/  ff,  %  t,  j; 

It  is  worthy  of  note  that  this  has  been  completely  derived 
by  the  principles  of  this  section  together  with  Gordan's  series. 

III.  Reducible  transvectants  in  general.  In  the  trans- 
vectants  studied  in  (I)  of  this  section,  e.g.  (<k_i,  /)',  the 
second  form  is  simple,  /=  a™,  of  the  first  degree.  It  is  pos- 
sible and  now  desirable  to  extend  those  methods  of  proving 
certain  transvectants  to  be  reducible  to  the  more  general  case 
where  both  forms  in  the  transvectants  are  monomials  in  other 
concomitants  of  lesser  degree. 

Consider  two  systems  of  binary  forms,  an  (J.)  system  and 
a  (i?)  s}rstem.     Let  the  forms  of  these  systems  be 

(yl)  :  Av  A2,  •••,  Ak,  of  orders  av  a3,  •••,  ak  respectively; 

and 

(i?)  :  Bv  By  •••,  Bt,  of  orders  bv  b.r  •  ••,  bt  respectively. 

Suppose  these  forms  expressed  in  the  Aronhold  symbolism 
and  let 

<f>  =  AfA*  •••  Jlj»,  yfr  =  B\*B**  •••  BfK 

Then  a  system  (C)  is  said  to  be  the  system  derived  by  trans- 
vection  from  '(A)  and  (B~)  when  it  includes  all  terms  in  all 
transvectants  of  fcfao-type 

(<£,   yjry  (133) 


REDUCTION  103 

Evidently  the  problem  of  reducibility  presents  itself  for 
analysis  immediately.     For  let 

$  =  per,  -^  =  pv, 

and  suppose  that,/  can  be  separated  into  two  integers, 

J-Ji+Jv 

such  that  the  trans vectants 

(pi  t*y\    (<*<  V)*> 

both  exist  and  are  different  from  zero.  Then  the  process 
employed  in  proving  formula  (132)  shows  directly  that 
(</>,  ■tyy  contains  terms  which  are  products  of  terms  of  (p,  /x)Ji 
and  terms  of  (<r,  v)1'*-,  that  is,  contains  reducible  terms. 

In  order  to  discover  what  transvectants  of  the  ( (7)  system 
contain  reducible  terms  we  employ  an  extension  of  the 
method  of  Paragraph  (I)  of  this  section.  This  may  be 
adequately  explained  in  connection  with  two  special  systems 

(A)=f,    (B)=i, 

where  f  is  a  cubic  and  i  is  a  quadratic.     Here 

(C)=|(<k  ^)fU(/*,p>-. 

Since  fa  must  not  contain  a  factor  of  order  "^j,  we  have 

3a— 3  <y<3«;  y  =  3  a,  3a  —  1,  3a  —  2. 
Also 

2/3-2<y<2/3;y  =  2A2/3-l. 

Consistent  with  these  conditions  we  have 

(/,  p,   (/,  p«    (/,  f )»,   (f\  p)*    (/*,  f)»    C/2,  p)6, 

(Z3,  f)7,  C/3,f)8,  CA  t8)9,  •••• 

Of  these,  (f2,  i2)4  contains  terms  of  the  product  (/,  i)2  (/,  i)2, 
that  is,  reducible  terms.  Also  (f\  i3)5  is  reducible  by  (/,  i)2 
(/,  t2)3.     In  the  same  way  (/3,  e4)7,  •••all  contain  reducible 


104  THE   THEORY   OF   INVARIANTS 

terms.  Hence  the  transvectants  of  ((7)  which  do  not  con- 
tain reducible  terms  are  six  in  number,  viz. 

/,  i,   (/,  i),   (J,  i)\   (/,  fy,   (/»    {3)6. 

The  reader  will  find  it  very  instructive  to  find  for  other  and 
more  complicated  (A~)  and  (2?)  systems  the  transvectants  of 
(C7)  which  do  not  contain  reducible  terms.  It  will  be  found 
that  the  irreducible  transvectants  are  in  all  cases  finite  in 
number.  This  will  be  proved  as  a  theorem  in  the  next 
chapter. 

SECTION  4.     SYZYGIES 

We  can  prove  that  m  is  a  superior  limit  to  the  number  of 
functionally  independent  invariants  and  covariants  of  a 
single  binary  form  /=  a"!  of  order  m.  The  totality  of  in- 
dependent relations  which  can  and  do  subsist  among  the 
quantities 

xv  xv  x'v  x'2,  a'i,  cti  (i=  0,  •  •-,  m),  Xr  X2,  fiv  /i2,  M—  (\\x) 

are  m  +  4  in  number.     These  are 

a^  =  <_i«;  (i  =  0,  •  -.,  m)  ;    x1  =  XrrJ  +  nrr'2,  x2  =  \x[  +  n2x'2  ; 

31=  X^  -  X2^r 

When  one  eliminates  from  these  relations  the  four  variables 
\v  X2,  fxv  /x2  there  result  at  most  m  relations.  This  is  the 
maximum  number  of  equations  which  can  exist  between 
a[,  at  (i  =  0,  •••,  m),  xv  x2,  x'v  x2,  and  M.  That  is,  if  a  greater 
number  of  relations  between  the  latter  quantities  are  as- 
sumed, extraneous  conditions,  not  implied  in  the  invariant 
problem,  are  imposed  upon  the  coefficients  and  variables. 
But  a  concomitant  relation 

<K«o>  •••'  a'»'  xv  4)  =  ^<K«(r  •"!  am>  vv  *2) 

is  an  equation  in  the  transformed  coefficients  and  variables, 
the  untransformed  coefficients  and  variables  and  M.     Hence 


EEDUCTION 


105 


there  cannot  be  more  than  m  algebraically  independent  con- 
comitants as  stated. 

Now  the  fundamental  system  of  a  cubic  contains  four  con- 
comitants which  are  such  that  no  one  of  them  is  a  rational 
integral  function  of  the  remaining  three.  The  present 
theory  shows,  however,  that  there  must  be  a  relation  be- 
tween the  four  which  will  give  one  as  a  function  of  the  other 
three  although  this  function  is  not  a  rational  integral  func- 
tion. Such  a  relation  is  called  a  syzygy  (Cay ley).  Since  the 
fundamental  system  of  a  quartic  contains  five  members  these 
must  also  be  connected  by  one  syzygy.  We  shall  discover 
that  the  fundamental  system  of  a  quintic  contains  twenty- 
three  members.  The  number  of  syzygies  for  a  form  of  high 
order  is  accordingly  very  large.  In  fact  it  is  possible  to  de- 
duce a  complete  set  of  syzygies  for  such  a  form  in  several  ways. 
There  is,  for  instance,  a  class  of  theorems  on  Jacobians  which 
furnishes  an  advantageous  method  of  constructing  syzygies. 
We  proceed  to  prove  these  theorems. 

I.  Theorem.  If  f,  g,  h  are  three  binary  forms,  of  respec- 
tive orders  n,  m,  p  all  greater  than  unity,  the  iterated  Jacobian 
((/,  g),  li)  is  reducible. 

The  three  series 


give  the  respective  results 


f    9 

li\ 

Ui 

/ 

f 

n    m    p 

p    n 

m 

.0     1     1J 

[o   1 

lj 

'  b 

'     A    /] 

n 

,    p     n 

.0 

1     1, 

((/,  A),  g)  + 


n  +  p 


1 


^  (f  a  )2g  - 


-i 


m  +  n 


5  (/.*»**■ 


100 


THE   THEORY  OF   INVARIANTS 


m  +  p  —  1 

m  +  n  —  z 


u  -\-p  —  I 

m  +  p  - 


o(*»/)Vi 


(<7,  fc)2/. 


We  add  these  equations  and  divide  through    by  2,  noting 
that  (/,  g)=-(g,f),  and  obtain 


n  —  m 


2(,»+n-2)  (134) 

This  formula  constitutes  the  proof  of  the  theorem.  It 
may  also  be  proved  readily  by  transvection  and  the  use  of 
reduction  identity  (101). 

II.  Theorem.  If  e  =  a";.  f=h%,  g  =  c%,  h  =  dqr  are  four 
binary  forms  of  orders  greater  than  unity,  then 

(135) 

We  first  prove  two  new  symbolical  identities.  By  an 
elementary  rule  for  expanding  determinants 


Hence 


"  1  ".> 


—  (ad))(bc)(ca). 


a\ 

1 


1"2 
hA 


VI 


b\ 
4 


d\     --dJA     d\ 

el       - 2  e2e\         gl 
/I        -2/a/!       fi 

=  2(ah)(hc){eaXde)(ef){fd) 

(adf     O)2     (a/)2 
=  (bd)2     (be)2     (bfy 

(cdy  (cey   (c/y 

In  this  identity  set  <?1=  —2^,  c2=xv  f\=  —x2,  f2  =  rr 


(136) 


REDUCTION 


107 


Then  (136)  gives  the  identity. 

(ad~)2  (ae)2  a| 

2(ab)(de)axbxdxex  =   (bdf  (bef  J2  . 

d%  el  0 
We  now  have 

(e,f)(g,  h)  =  (ab')(cd~)a™-^bnl-^c»-'Ld(1-1 

(ac)2     (ar?)2     a2 

= i  a^r^r^r2  (6c)2  (6(^2  5-2- 

<?l  d%  0 

by  (137).     Expanding  the  right-hand  side  we  have  formula 
(135)  immediately. 

III.  Theorem.  The  square  of  a  Jacobian  J  =  (/,  g)  is  given 
by  the  formula 

-2J*  =  (ff)Y  +  (ff,  gyp  -  2  (/,  gYfg.  (138) 

This  follows  directly  from  (135)  by  the  replacements 

"     fif     9,  9 '  ^&=kg  //• 

IV.  Syzygies  for  the  cubic  and  quartic  forms.  In  formula 
(138)  let  us  make  the  replacements  J  =  Q,  f=f,  g  =  A, 
where  /  is  a  cubic,  A  is  its  Hessian,  and  Q  is  the  Jacobian 
(/,  A).     Then  by  Table  I 

AS'~2^2  +  A3+JR/2=0.  (139) 

This  is  the  required  syzygy  connecting  the  members  of  the 
fundamental  system  of  the  cubic. 

Next  let  /,  -H",  T,  i,  J  be  the  fundamental  system  of  a 
quartic  /.  Then,  since  T  is  a  Jacobian,  let  J=  T,  f  =  f, 
g  =  Hm  (138),  and  we  have 

-2F=F_  2(/,  HyfH+  (H,  H  )2/2. 
But  by  Table  II 

(fHy  =  lif,  (E,Hy=\(ZJf-iH}. 


108  THE   THEORY   OF   INVARIANTS 

Hence  we  obtain 

S  =  2  T2  +  H*  -  i  ipH  +  J  Jf  3  =  0.  ( 140) 

This  is  the  syzygy  connecting  the  members  of  the  funda- 
mental system  of  the  quartic. 

Of  the  twenty-three  members  of  a  system  of  the  quintic 
nine  are  expressible  as  Jacobians  (cf.  Table  IV,  Chap.  VI). 
If  these  are  combined  in  pairs  and  substituted  in  (135),  and 
substituted  singly  in  (138),  there  result  45  syzygies  of  the  type 
just  derived.  For  references  on  this  subject  the  reader  may 
consult  Meyer's  "  Bericht  ueber  den  gegenwiirtigen  Stand 
der  Invariantentheorie "  in  the  Jahresbericht  der  Deutschen 
Mathematiker-Vereinigung  for  1890-91. 

V.  Syzygies  derived  from  canonical  forms.  We  shall  prove 
that  the  binary  cubic  form, 

/=  aQx\  +  3  axx^x2  +  3  a2xxx\  +  azx\, 

may  be  reduced  to  the  form, 

f=x3+  r3. 

by  a  linear  transformation  with  non-vanishing  modulus.  In 
general  a  binary  quantic /of  order  m  has  m  +  1  coefficients. 
If  it  is  transformed  by 

T:  x1  =  \rr[  +  fi^  x2  =  \x{  +  fx2z'2, 

four  new  quantities  \v  /jlv  X2,  /x2  are  involved  in  the  coeffi- 
cients of/'.  Hence  no  binary  form  of  order  m  with  less 
than  m  —  3  arbitrary  coefficients  can  be  the  transformed  of 
a  general  quantic  of  order  m  by  a  linear  transformation. 
Any  quantic  of  order  m  having  just  m  —  3  arbitrary  quanti- 
ties involved  in  its  coefficients  and  which  can  be  proved  to 
be  the  transformed  of  the  general  form  /  by  a  linear  trans- 
formation of  non-vanishing  modulus  is  called  a  canonical 
form  of/.  We  proceed  to  reduce  the  cubic  form  /  to  the 
canonical  form  Xz  +  Vs.  Assume 
f=a0ri+  •..  =p1(z1  +  a1x2y+p2(x1  +  a2z2y=X3+Y3.     (140!) 


REDUCTION 


109 


This  requires  that/ be  transformable  into  its  canonical  form 
by  the  inverse  of  the  transformations 

a        -r-  1  1  _  1  1 

o  :  X.  =  p\xx  +  p\a1x2,  I  =  p\x1  +  pla2x2. 

We  must  now  show  that  pv  p2,  av  «2  may  actually  be  de- 
termined, and  that  the  determination  is  unique.  Equating 
coefficients  in  (140^  we  have 

p1+p2  =  a0, 

ttiPi  +  HV%  =  av 

aiPi  +  a2p2  =  av 
a\p1  +  a\p2  =  a3. 

Hence  the  following  matrix,  M,  must  be  of  rank  2 

1 


(140,) 


M= 


Of 

4 


From  31=  0  result 
1        a. 


4  =o, 

a0\ 


=  0. 


Expanding  the  determinants  we  have 

Pa0  +  (?«!  +  Ra2  =  0, 
Pax  +  Qa2  +  Ra3  =  0. 
Also,  evidently 

P  +  Qa,  +Rd}  =  0     (i  =  l,  2). 

Therefore  our  conditions  will  all  be  consistent  if  av  «2  are 
determined  as  the  roots,  £x  -s- 12,  of 


\±  = 


o  l       ' 

aj       a2       <^3  =  0. 

ii  -fA  si 

This  latter  determinant  is  evidently  the  Hessian  of/,  divided 
by  2.      Thus  the  complete  reduction  of  /to  its  canonical  form 


110  THE   THEORY   OF   IX VARIANTS 

is  accomplished  by  solving  its  Hessian  covariant  for  the 
roots  av  «2,  and  then  solving  the  first  two  equations  of  (1402) 
for  pv  p2.  The  inverse  of  S  will  then  transform  /  into 
X3  +  Ys.     The  determinant  of  S  is 

&  =  (Pl  'Pz)    («2~«l)' 

and  2)  =£  0  unless  the  Hessian  has  equal  roots.  Thus  the 
necessary  and  sufficient  condition  in  order  that  the  canonical 
reduction  be  possible  is  that  the  discriminant  of  the  Hessian 
(which  is  also  the  discriminant,  R,  of  the  cubic/)  should  not 
vanish.     If  R  =  0,  a  canonical  form  of/ is  evidently  X-Y. 

Among  the  problems  that  can  be  solved  by  means  of  the 
canonical  form  are,  (a)  the  determination  of  the  roots  of  the 
cubic  /=  0  from 

x3+y3  =  (X+  Y)(x+<or)(x+^Y^ 

co  being  an  imaginary  cube  root  of  unit}-,  and  (£)  the  deter- 
mination of  the  syzygy  among  the  concomitants  of  /  We 
now  solve  problem  (£).  From  Table  I,  by  substituting 
a0  =  a3  =  1,  a1  =  a2  =  0,  we  have  the  fundamental  system  of 
the  canonical  form : 

x3+r3,  2i7,  x3-r3,  -2. 

Now  we  may  regard  the  original  form/  to  be  the  transformed 
form  of  X3  +  Y3  under  S.  Hence,  since  the  modulus  of  S 
is  D,  we  have  the  four  invariant  relations 

f=X*  +  T3, 
A  =  2  D*XY, 
Q  =  DHXS-  F3), 
R  =  -  D«  .  2. 

It  is  an  easy  process  to  eliminate  D,  X,  Y  from  these  four 
equations.     The  result  is  the  required  syzygy : 

PR+  2  £2  +  A3  =  0. 


EEDUCTION 


111 


A  general  binary  qnartic  can  be  reduced  to  the  canonical 
form  (Cayley) 

X4+  Y*+  QmX2Y2; 

a  ternary  cubic  to  the  form  (Hesse) 

Xs  +  r3  +  Z*  +  6  mXYZ. 

An  elegant  reduction  of  the  binary  quartic  to  its  canonical 
form  may  be  obtained  by  means  of  the.provectant  operators 
of  Chapter  III,  §  1,  V.     We  observe  that  we  are  to  have 

identically 

/=  (a0,  av  .-.,  a,  $  xv  x2Y  =  X\  +  X\  +  6  mX\X% 
where  Xv  X2  are  linear  in  xv  x2  ; 

Xx  =  a1x1  +  «2.r2,  X2  =  /S^j  +  /32.r2. 
Let  the  quadratic  XtX2  be  ^  =  (A0,  ^4r  A2\xv  .r2)2.     Then 

5?  ■  XJ  =  (A0,  Av  A&4-*  -T-yXf  =  °    O"  =  li  2>- 


d.r. 


5^! 


6  m5?  •  XfXl  =  12  •  2(4  J.^  -  A\)mXxX^  =  12  X  XXX2. 

Equating  the  coefficients  of  a;f,  sr^,  #|  in  the  first  equation 
above,  after  operating  on  both  sides  by  dq,  we  now  have 

A0a2  -  Axax  +  A2a0  =  \A0, 
A0a3  —  Axa2  +  A2ax  =  £  \AV 
AqU^  —  Axaz  +  ^L2«2  =  \^42. 

Forming  the  eliminant  of  these  we  have  an  equation  which 
determines  X,  and  therefore  ra,  in  terms  of  the  coefficients  of 
the  original  quartic/.      This  eliminant  is 

X 


"1  l*2 


a2  -  X 


0, 


or,  after  expanding  it, 


112  THE   THEORY  OF   INVARIANTS 

where  i,  J  are  the  invariants  of  the  quartic  /  determined  in 
Chapter  III,  §  1,  V.  It  follows  that  the  proposed  reduction 
of  f  to  its  canonical  form  can  be  made  in  three  ways. 

A  problem  which  was  studied  by  Sylvester,*  the  reduction 
of  the  binaiy  sextic  to  the  form 

X\  +  X\  +  X%  +  30  tnX\X\X% 

has  been  completely  solved  very  recently  by  E.  K.  Wakeford.f 

SECTION   5.     IIILBERT'S   THEOREM 

We  shall  now  prove  a  very  extraordinary  theorem  due  to 
Hilbert  on  the  reduction  of  systems  of  qualities,  which  is  in 
many  ways  closely  connected  with  the  theoiy  of  syzygies. 
The  proof  here  given  is  by  Gordan.  The  original  proof  of 
Hilbert  may  be  consulted  in  his  memoir  in  the  Mathematische 
Annalen,  volume  36. 

I.  Theorem.  If  a  homogeneous  algebraical  function  of  any 
number  of  variables  be  formed  according  to  any  definite  laws, 
then,  although  there  mag  be  an  infinite  number  of  functions  F 
satisfying  the  conditions  laid  down,  nevertheless  a* finite  numln  r 
Fv  Fv  ••.  Fr  can  always  be  found  so  that  any  other  F  can  be 
written  in  the  form 

F=AlF1  +  A2F2  +  •••  +ArFr. 

where  the  As  are  homogeneous  integral  functions  of  the  variables 
but  do  not  necessarily  satisfy  the  conditions  for  the  F's. 

An  illustration  of  the  theorem  is  the  particular  theorem 
that  the  equation  of  any  curve  which  passes  through  the  in- 
tersections of  two  curves  Fx  =  0,  F2  =  0  is  of  the  form 

F  =  AXFX  -I-  A2F2  =  0. 

Here  the  law  according  to  which  the  F's  are  constructed  is 
that  the  corresponding  curve  shall  pass  through  the  stated 

*  Cambridge  and  Dublin  MathematicalJournal,  vol.  6  (18ol),  p.  293. 
t  Messenger  of  Mathematics,  vol.  43  (1913-14),  p.  25. 


REDUCTION  113 

intersections.  There  are  an  infinite  number  of  functions  sat- 
isfying this  law,  all  expressible  as  above,  where  Av  A2  are 
homogeneous  in  xv  x2,  :r3  but  do  not,  as  a  rule,  represent 
curves  passing  through  the  intersections. 

We  first  prove  a  lemma  on  monomials  in  <n  variables. 

Lemma.  If  a  monomial  xfafy  •••  #*«,  where  the  lis  are 
positive  integers,  be  formed  so  that  the  exponents  kv  •  •-,  kn 
satisfy  prescribed  conditions,  then,  although  the  number  of 
products  satisfying  the  given  conditions  may  be  infinite,  never- 
theless a  finite  number  of  them  can  be  chosen  so  that  every  other 
is  divisible  by  one  at  least  of  this  finite  number. 

To  first  illustrate  this  lemma  suppose  that  the  prescribed 
conditions  are 

2  Jc,  +  3  K  -  h»  -  k.  =  0, 

1  2  6  4  (141) 

k1  +  k4  =  k2  +  k3. 
Then  monomials  satisfying  these  conditions  are 

•t'littjit   Ott    I  ^         rt   lXOi(  J^  .t    (yil    ty.l      p  .(    1  .(    p'*    •>•'      M 

and  all  are  divisible  by  at  least  one  of  the  set  x\x%c±,  x2xsx^. 

Now  if  n  =  1,  the  truth  of  the  lemma  is  self-evident.  For 
£fclrof  any  set  of  positive  powers  of  one  variable  are  divisible 
by  that  power  which  has  the  least  exponent.  Proving  by 
induction,  assume  that  the  lemma  is  true  for  monomials  of 
n  —  1  letters  and  prove  it  true  for  n  letters. 

Let  K=  x\'x2*  •••  xknn  be  a  representative  monomial  of  the  set 
given  by  the  prescribed  conditions  and  let  P  =  x^x?f  •••  x%n  be 
a  specific  pxod-uet-  of  the  set.  If  _ZT is  not  divisible  by  P,  one 
of  the  numbers  k  must  be  less  than  the  corresponding  num- 
ber a.     Let  kr  <  ar.     Then  kr  has  one  of  the  series  of  values 

0,  1,  2,  ....  ar-  1. 

t4w4-is,-  the  number  of  ways  that  this  can  occur  for  a  single 
exponent  is  finite  and  equal  to 

iVr=  ax+  a%  +  •••  +  an. 


114  THE   THEORY   OF   INVARIANTS 

The  cases  are 

kx  equals  one  of  the  series  0,1,  •••,  ax  —  1;   (ax  cases), 

k2  equals  one  of  the  series  0,  1,  •••,  a2  —  1 ;   (a2  cases),      (142) 

etc. 

Now  let  kr  =  m  and  suppose  this  to  be  case  number  p  of  (142). 
Then  the  n  —  1  remaining  exponents  kv  kv  •••,  kr_v  kr+r  •  ••, 
kn  satisfy  definite  conditions  which  could  be  obtained  by 
making  kT  =  m  in  the  original  conditions.      Let 

Kp  =  x\lX%  •  ■  ■  X™  ■  •  •  X*n  =  xfK'v 

be  a  monomial  of  the  system  for  which  kT  =  m.  Then  Kp 
contains  only  n  —  1  letters  and  its  exponents  satisfy  definite 
conditions  which  are  such  that  x'"K'p  satisfies  the  original 
conditions.  Hence  by  hypothesis  a  finite  number  of  mono- 
mials of  the  type  K p,  say, 

exist  such  that  all  monomials  K'p  are  divisible  by  at  least  one 
L.  Hence  Kp  =  x"lK'p  is  divisible  by  at  least  one  X,  and  so 
by  at  least  one  of  the  monomials 

Mp  =  x?Ly  Mf  =  x?ll  ....  M#J  =  a?z' 

Also  all  of  the  latter  set  of  monomials  belong  to  the  orig- 
inal system.  Thus  in  the  case  number  p  in  (142)  K  is 
divisible  by  one  of  the  monomials 

Now  suppose  that  if  is  not  divisible  by  P.  Then  one  of  the 
cases  (142)  certainly  arises  and  so  K  is  always  divisible  by 
one  of  the  pr©4uets 

m  cist'  by  P.  Hence  if  the  lemma  holds  true  for  monomials 
in  n  —  1  letters,  it  holds  true  for  n  letters,  and  is  true  univer- 
sally. 


KEDUCTION  115 

We  now  proceed  to  the  proof  of  the  main  theorem.  Let 
the  variables  be  xv  •  ••,  xn  and  let  F  he  a  typical  function  of 
the  system  described  in  the  theorem.  Construct  an  auxiliary 
system  of  functions  77  of  the  same  variables  under  the  law 
that  a  function  is  an  77  function  when  it  can  be  written  in  the 
form 

77  =  ZAF  (143) 

where  the  A's  are  integral  functions  rendering  ?/  homoge- 
neous, but  not  otherwise  restricted  except  in  that  the  number 
of  terms  in  77  must  be  finite. 

Evidently  the  class  of  77  functions  is  closed  with  respect  to 
linear  operations.     That  is, 

ZBV  =  BlVl  +  B,v,  +  •••  =  ZBAF  =  lA'F 

is  also  an  77  function.  Consider  now  a  typical  77  function. 
Let  its  terms  be  ordered  in  a  normal  order.  The  terms  will 
be  denned  to  be  in  normal  order  if  the  terms  -ef  an}'  pair,  t 

S  =  x?x%  •  •  •  a#s   T  =  a^aji  •  ■  •  a£», 

are  ordered  so  that  if  the  exponents  a,  b  of  S  and  T  are  read 
simultaneously  from  left  to  right  the  term  first  to  show  an 
exponent  less  than  the  exponent  in  the  corresponding  posi- 
tion in  the  other  term  occurs  farthest  to  the  right.  If  the 
normal  order  of  S,  T  is  (#,  T),  then  T  is  said  to  be  of  lower 
rank  than  S.  That  is,  the  terms  of  77  are  assumed  to  be 
arranged  according  to  descending  rank  and  there  is  a  term 
of  highest  and  one  of  lowest  rank.  By  hypothesis  the  77 
functions  are  formed  according  to  definite  laws,  and  hence 
their  first  terms  satisfy  definite  laws  relating  to  their  expo- 
nents. B}r  the  lemma  just  proved  we  can  choose  a  finite 
-n-mnber  of  r\  functions,  77 r  772,  •••,  77^  such  that  the  first  term 
of  any  other  77  is  divisible  by  the  first  term  of  at  least  one 
of  this  number.  Let  the  first  term  of  a  definite  77  be 
divisible  by  the  first  term  of  77mi  and  let  the  quotient  be  Pv 


11(3  THE   THEORY   OF   INVARIANTS 

Then  n  —  Pxrimi  is  an  n  function,  and  its  first  term  is  of 
lower  rank  than  the  first  term  of  ?;.     Let  this  be  denoted  by 

Suppose  next  that  the  first  term  of  77(1)  is  divisible  by  r;m, ; 

thus. 

and  the  first  term  of  v(2)  is  of  lower  rank  than  that  of  tja). 
Continuing,  we  obtain 

Then  the  first  terms  of  the  ordered  set 

77,  va>,  v®\  ...,  V',  ••• 

are  in  normal  order,  and  since  there  is  a  term  of  lowest  rank 
in  77  we  must  have  for  some  value  of  r 

V0"  =  Pr+lVmr+1- 

That  is,  we  must  eventually  reach  a  point  where  there  is  no 
■q  function  77(,'+1)  of  the  same  order  as  n  and  whose  hist 
term  is  of  lower  rank  than  the  first  term  of  w(r) .     Hence 

V  =  Pi>Ux  +  PzVm,  +  ■  •  •  +  Pl+iVmr+1  ( 144) 

and  all  n's  on  the  right-hand  side  are  members  of  a  definite 
finite  set 

Vr  1h<  "••  V 
But  by  the  original  theorem  and  (143),  every  F  is  itself  an 
rj  function.     Hence  by  (144  ) 

F=  AXFX  +  A0F2  +  •••  +  ArFr.  (145) 

where  Ft(i  =  l,  •••,  r)  are  the  F  functions  involved  linearly 
in  ijv  ?72,  •••,  np.     This  proves  the  theorem. 

II.  Linear  Diophantine  equations.  If  the  conditions  im- 
posed upon  the  exponents  k  consist  of  a  set  of  linear  Dio- 
phantine equations  like  (141),  the  lemma  proved  above  shows 
that  there  exists  a  set  of  solutions  finite  i)i   number  by  mean* 


REDUCTION  117 

of  which  any  other  solution  can  be  reduced.     That  is,  this  fact 
follows  as  an  evident  corollary. 

Let  us  treat  this  question  in  somewhat  fuller  detail  by  a 
direct  analysis  of  the  solutions  of  equations  (141).  The 
second  member  of  this  pair  has  the  solutions 


fc\i 

K 

£g, 

h, 

(1) 

0 

0 

1 

1 

(2) 

0 

l 

0 

1 

(3) 

1 

0 

1 

0 

(4) 

1 

1 

0 

0 

(5) 

1 

1 

1 

1 

(6) 

2 

1 

1 

0 

Of  these  the  fifth  is  obtained  by  adding  the  first  and  the 
fourth ;  the  sixth  is  reducible  as  the  sum  of  the  third  and 
the  fourth,  and  so  on.  The  sum  or  difference  of  any  two 
solutions  of  any  such  linear  Diophantine  equation  is  evi- 
dently again  a  solution.  Thus  solutions  (1),  (2),  (3),  (4) 
of  kx  +  ki  =  k2  -f-  k3  form  the  complete  set  of  irreducible 
solutions.  Moreover,  combining  these,  we  see  at  once  that 
the  general  solution  is 

(I)  k1  =  x  +  y,  k2=  x  +  z,  k3  =  y  -\-  w,  k4  =z  +  w. 

Now  substitute  these  values  in  the  first  equation  of  (141) 

2  kx  +  3  \  -  kz  -  k±  =  0. 
There  results 

5x  +  y  -\-  2z=2w. 

By  the  trial  method  illustrated  above  we  find  that  the  irre- 
ducible solutions  of  the  latter  are 

a;  =  2,  w  =  5;  y  =  %  w  =\\  z  =  1,  w  =  1 ;  x  =1,  y  =  1,  w  =  3, 

where  the  letters  not  occurring  are  understood  to  be  zero. 
The  general  solution  is  here 

(II)  x=2a  +  d,  y  =  2b  +  d,  z  =  c,  w  =  5a  +  b  +  c+3d, 


118  THE   THEORY  OF   INVARIANTS 

and  if  these  be  substituted  in  (I)  we  have 

&!  =  2  a  +  2 1  +2d 

k.2  =  2  a  +     c  +     c? 

k3  =  5  a  +  3  !*  +     c  +  4  d 
&4  =  5  a  +     ft  +  2  e  +  3  d 

Therefore  the  only  possible    irreducible  simultaneous  solu- 
tions of  (111)  are 


*n 

A?2? 

*3, 

h 

(1) 

2 

2 

5 

5 

(2) 

0 

0 

3 

1 

(3) 

0 

1 

1 

2 

(4) 

2 

1 

4 

3 

But  the  first  is  the  sum  of  solutions  (3)  and  (4)  ;  and  (4)  is 
the  sum  of  (2)  and  (3).  Hence  (2)  and  (3)  form  the  com- 
plete set  of  irreducible  solutions  referred  to  in  the  corollary. 
The  general  solution  of  the  pair  is 

k1==2  a,  Jc2  =  &  k3  =  3  «  +  yS,  *4  =  «  +  2  j3. 

The  corollary  may  now  be  stated  thus: 

Corollary.  Every  simultaneous  set  of  linear  homogeneous 
Diophantine  equations  possesses  a  set  of  irreducible  solutions, 
finite  in  number.  A  direct  proof  without  reference  to  the 
present  lemma  is  not  difficult.*  Applied  to  the  given  illus- 
tration of  the  above  lemma  on  monomials  the  above  analysis 
shows  that  if  the  prescribed  conditions  on  the  exponents  are 
given  by  (141)  then  the  complete  system  of  monomials  is 
given  by 

™2a,,j3„3a+^     a+2/3 
•'  1  *(  2    3        •*  4        ' 

where  a  and  /3  range  through  all  positive  integral  values 
independently.  Every  monomial  of  the  system  is  divisible 
by  at  least  one  of  the  set 

*  Elliott,  Algebra  of  Qualities,  Chapter  IX. 


REDUCTION  119 

which  corresponds  to  the  irreducible  solutions  of  the  pair 
(141). 

III.  Finiteness  of  a  system  of  syzygies.  A  syzygy  S 
among  the  members  of  a  fundamental  system  of  concomitants 
of  a  form  (cf.  (140))/, 

Iv  Iv  ■  ••,  ZM,  Kv  ... 

is  a  polynomial  in  the  i"'s  formed  according  to  the  law  that 
it  will  vanish  identically  when  the  J's  are  expressed  ex- 
plicity  in  terms  of  the  coefficients  and  variables  of  /.  The 
totality  of  syzygies,  therefore,  is  a  system  of  polynomials 
(in  the  invariants  _T)  to  which  Hilbert's  theorem  applies.  It 
therefore  follows  at  once  that  there  exists  a  finite  number  of 
syzygies, 

1'       2'    *"1       vi 

such  that  any  other  syzygy  8  is  expressible  in  the  form 

8=  0&  +  C2S2  +  •••  +  0V8V.  (146) 

Moreover  the  C's,  being  also  polynomials  in  the  Z's  are 
themselves  invariants  of/.     Hence 

Theorem.  The  number  of  irreducible  syzygies  among  the 
concomitants  of  a  form  f  is  finite,  in  the  sense  indicated  by 
equation  (146). 

SECTION   6.     JORDAN'S   LEMMA 
Many  reduction  problems  in  the  theory  of  forms  depend  for 
their  solution  upon  a  lemma  due  to  Jordan  which  may  be 
stated  as  follows  : 

Lemma.  If  ux  +  u%  +  u3  =  0,  then  any  product  of  powers  of 
uv  u2,  u3  of  order  n  can  be  expressed  linearly  in  terms  of  such 
products  as  contain  one  exponent  equal  to  or  greater  than  f  n. 

We  shall  obtain  this  result  as  a  special  case  of  a  consider- 
ably more  general  result  embodied  in  a  theorem  on  the 
representation  of  a  binary  form  in  terms  of  other  binary 
forms. 


120 


THE   THEORY   OF   INVARIANTS 


I.  Theorem.  If  ax,  bx,  cx,  •••  are  r  distinct  linear  forms,  and 
A,  B,  O,  •••  are  binary  forms  of  the  respective  orders  a,  /3,  7.  ••• 
where 

a  +  £+7  +  ...  =n-r  +  l, 

then  any  binary  form  f of  order  n  can  be  expressed  in  the  form 

f=  a"x-aA  +  bnx-»B+  0^yC+  -.., 
and  the  expression  is  unique. 

As  an  explicit  illustration  of  this  theorem  we  cite  the 
case  n  =  3,  r  =  2.     Then  «  +  /3  =  2,  «  =  /3  =  1. 

/=  a1(.Pooxi  +Povh)  +  hl  (PvFi  +Pnx2)-  (147) 

Since  /,  a  binary  cubic,  contains  four  coefficients  it  is  evi- 
dent that  this  relation  (147)  gives  four  linear  nonhomo- 
geneous  equations  for  the  determination  of  the  four  unknowns 
Poo*  Pov  Pw>  Piv  Thus  the  theorem  is  true  for  this  case  pro- 
vided the  determinant  representing  the  consistency  of  these 
linear  equations  does  not  vanish.  Let  ar  =  a1x1  +  a0x2, 
bx  =  byX-y  +  b2x2,  and  D  =  axb2  —  a2br  Then  the  aforesaid 
determinant  is 


«? 

0 

H 

0 

2  axa2 

a\ 

2  b,b2 

n 

a\ 

2  ax«3 

H 

2*A 

0 

a\ 

0 

b\ 

This  equals  .D4,  and  D  ^  0  on  account  of  the  hypothesis 
that  a,,  and  bx  are  distinct.  Hence  the  theorem  is  here  true. 
In  addition  to  this  we  can  solve  for  the  ptj  and  thus  deter- 
mine A.  B  explicitly.  In  the  general  case  the  number  of 
unknown  coefficients  on  the  right  is 

a  +  /3  +  7  +  •••  +r  =  n  +  l. 

Hence  the  theorem  itself  may  be  proved  by  constructing  the 
corresponding  consistency  determinant  in  the  general  case  ;  * 
but  it  is  perhaps  more  instructive  to  proceed  as  follows : 

*  Cf.  Transactions  Amer.  Math.  Society,  Vol.  15  (1914),  p.  SO. 


REDUCTION  121 

It  is  impossible  to  find  r  binary  forms  A,  B,  C,  •••  of  orders 

a,  /?,  7,  •••  where 

«  +  /3  +  7+  ...  =  n  —  r  +  1, 

such  that,  identically, 

E=  an~aA  +  b%-?B  +  c'^0+  •••  =0. 

In  fact  suppose  that  such  an  identity  exists.  Then  operate 
upon  both  sides  of  this  relation  a  +  1  times  with 

A  =  a2  z (h  t~     O*  =  <hx\  +  aix%>- 

Let  gx  be  any  form  of  order  n  and  take  a2  =  0.     Then 

Aa+V"  =  &Oi  •  ^)a+Vra_I 

where  the  &'s  are  numerical.  Hence  Aa+1g"  cannot  vanish 
identically  in  case  a2  =  0,  and  therefore  not  in  the  general 
case  a2=£  0,  except  when  the  last  n  —  a  coefficients  of  g"  vanish: 
that  is,  unless  g%  contains  an~a  as  a  factor.     Hence 

where  B ',  C  are  of  orders  /3,  7,  •••  respectively.  Now 
Aa+1B  is  an  expression  of  the  same  type  as  E,  with  r  changed 
into  r  —  1  and  n  into  n  —  a  —  1,  as  is  verified  by  the  equation 

/3  +  7  +   ...  =  (h  —  a—  1)  —  (r—  1)  +  1  =  ra  —  r  +  1  —  a. 

Thus  if  there  is  no  such  relation  as  E=Q  for  r—1  linear 
forms  ax,  bx,  •••,  there  certainly  are  none  for  r  linear  forms. 
But  there  is  no  relation  for  one  form  (r  =  1)  save  in  the 
vacuous  case  (naturally  excluded)  where  A  vanishes  identi- 
cally. Hence  by  induction  the  theorem  is  true  for  all  values 
of  r. 

Now  a  count  of  coefficients  shows  at  once  that  any  binary 
form /of  order  n  can  be  expressed  linearly  in  terms  of  n  +  1 


12-2  THE   THEORY   OF   INVARIANTS 

binary  forms  of  the  same  order.  Hence  /  is  expressible  in 
the  form 

/=  anr-aA  +  b''-^B  +  e%-ty  +  .... 

That  the  expression  is  unique  is  evident.  For  if  two  such 
were  possible,  their  difference  would  be  an  identically  vanish- 
ing expression  of  the  type  i?=  0,  and,  as  just  proved,  none 
such  exist.     This  proves  the  theorem. 

II.  Jordan's  lemma.  Proceeding  to  the  proof  of  the 
lemma,  let  u3  =  —  (w2  +  i/2),  supposing  that  uv  u2  replace  the 
variables  in  the  Theorem  I  just  proved.  Then  w3,  uv  u2  are 
three  linear  forms  and  the  Theorem  I  applies  with  r=3, 
a  +  ft  +  y  =  n  —  2.  Hence  any  homogeneous  expression  /  in 
uv  uv  us  can  be  expressed  in  the  form 

ufr-M  +  u'r^B  +  u%-yC, 

or,  if  we  make  the  interchanges 

n  —  a     n  —  ft     n  —  y 

X  (A  v 

in  the  form  u\A  +  u%B  +  i%C,  (148) 

where  \  +  \x  +  v  =  2  n  +  2.  (149) 

Again  integers  X,  /*,  v  may  always  be  chosen  such  that  (14'.') 
is  satisfied  and 

X  ^  -|  n,  \x  >  |  w,  v  >  |  n. 

Hence  Jordan's  lemma  is  proved. 

A  case  of  three  linear  forms  ut  for  which  Wj  +  «2  +  m3  =  0 
is  furnished  by  the  identity 

(ab^)cx  +  (bc)ax  +  (ca)bx  =  0. 

If  we  express  A  in  (118)  in  terms  of  uv  u2  by  means  of 
ux  +  u2  +  u3  =  0,  B  in  terms  of  w2,  w3,  and  (7  in  terms  of  ^3.  ur 
we  have  the  conclusion  that  any  product  of  order  n  of  (^)cx, 
(6c)ax,  (ca)bx  can  be  expressed  linearly  in  terms  of 


REDUCTION  123 

(abyc%,  (aby'-Hbe)^1-^,  (aby^(bcy<»-*a%  •  •-, 
(aby(bcy-xcAraH-\ 

(bcya%  (bey-^ca)^-^,,,  (bcy-%caya^~2b%  •••, 

(bey(ea)*-ilaftb%-ii,  (150) 

(cayb%  (cay~\ab')b^ex,  (eay-%abyb%-2c%  •••, 
(eay(aby-vbvxcnx-v, 

where  X  >  -§-  n,  fi  >  |  n,  v  ^>  f  w. 

It  should  be  carefully  noted  for  future  reference  that  this 
monomial  of  order  n  in  the  three  expressions  (ot6)cf,  (bc)a^ 
(ccL)br  is  thus  expressed  linearly  in  terms  of  symbolical 
products  in  which  there  is  always  present  a  power  of  a  deter- 
minant of  type  (a5)  equal  to  or  greater  than  %n.  The 
weight  of  the  coefficient  of  the  leading  term  of  a  covariant  is 
equal  to  the  number  of  determinant  factors  of  the  type  (a&) 
in  its  symbolical  expression.  Therefore  (150)  shows  that  if 
this  weight  w  of  a  covariant  of/ does  not  exceed  the  order  of 
the  form/ all  covariants  having  leading  coefficients  of  weight 
w  and  degree  3  can  be  expressed  linearly  in  terms  of  those  of 
grade  not  less  than  |  w.  The  same  conclusion  is  easily  shown 
to  hold  for  covariants  of  arbitrary  weight. 

SECTION   7.     GRADE 

The  process  of  finding  fundamental  systems  by  passing 
step  by  step  from  those  members  of  one  degree  to  those  of  the 
next  higher  degree,  illustrated  in  Section  3  of  this  chapter, 
although  capable  of  being  applied  successfully  to  the  forms 
of  the  first  four  orders,  fails  for  the  higher  orders  on  account 
of  its  complexity.  In  fact  the  fundamental  system  of  the 
quintic  contains  an  invariant  of  degree  18  and  consequently 
there  would  be  at  least  eighteen  successive  steps  in  the  process. 
As  a  proof  of  the  flniteness  of  the  fundamental  system  of  a 
form  of  order  n  the  process  fails  for  the  same  reason.    That  is, 


124  THE   THEORY   OF   INVARIANTS 

it  is  impossible  to  tell  whether  the  system  will  be  found  after 
a  finite  number  of  steps  or  not. 

In  the  next  chapter  we  shall  develop  an  analogous  process 
in  which  it  is  proved  that  the  fundamental  s}rstem  will  result 
after  a  finite  number  of  steps.  This  is  a  process  of  passing 
from  the  members  of  a  given  grade  to  those  of  the  next 
higher  grade. 

I.  Definition.  The  highest  index  of  any  determinant  factor 
of  the  type  («6)  in  a  monomial  symbolical  concomitant  is 
called  the  grade  of  that  concomitant.  Thus  (a6)4(ac)26J<?*  is 
of  grade  4.  The  terms  of  covariants  (84).  (87)  are  each  of 
grade  2. 

Whereas  there  is  no  upper  limit  to  the  degree  of  a  con- 
comitant of  a  form /of  order  n,  it  is  evident  that  the  maximum 
grade  is  n  by  the  theory  of  the  Aronhold  symbolism.  Hence 
if  we  can  find  a  method  of  passing  from  all  members  of  the 
fundamental  system  of/ of  one  grade  to  all  those  of  the  next 
higher  grade,  this  will  prove  the  finiteness  of  the  system, 
since  there  would  only  be  a  finite  number  of  steps  in  this 
process.  This  is  the  plan  of  the  proof  of  Gordan's  theorem 
in  the  next  chapter. 

II.  Theorem.  Every  covariant  of  a  single  form  f  of  odd 
grade  2X  —  1  can  be  transformed  into  an  equivalent  covariant  of 
the  next  higher  even  grade  2  X. 

We  prove,  more  explicitly,  that  if  a  symbolical  product 
contains  a  factor  (a5)2A_1  it  can  be  transformed  so  as  to  be 
expressed  in  terms  of  products  each  containing  the  factor 
(a6)2A.  Let  A  be  the  product.  Then  by  the  principles  of 
Section  2  A  is  a  term  of 

Hence  by  Theorem  III  of  Section  2. 


-f-^A'((rt^)2A-1«r1"2^rl"->A.  <£V.  a51) 


REDUCTION  125 

where  7'  <  7  and  ^>  is  a  concomitant  derived  from  </>  by  con- 
volution, K  being  numerical.  Now  the  symbols  are  equiva- 
lent.     Hence 

,],  =  (aJ)2*-1aj+1-2*^+1-2*  =  -  (aby^a^-^b^-'^  =  0. 

Hence  all  transvectants  on  the  right-hand  side  of  (151),  in 
which  no  convolution  in  -v/r  occurs,  vanish.  All  remaining 
terms  contain  the  symbolical  factor  (a6)2A,  which  was  to  be 
proved. 

Definition.  A  terminology  borrowed  from  the  theory 
of  numbers  will  now  be  introduced.  A  symbolical  product, 
A,  which  contains  the  factor  (a5)r  is  said  to  be  congruent  to 
zero  modulo  (a£)r; 

A  =  0  (mod  («&)'). 

Thus  the  covariant  (84) 

0  =  ^(a5)2(5a)2a|aa  4-  §  (ab~)\aa)(ba)axbxax 

gives  C=  ^(ab^2Qaa)(ba)axbxax(mod  (6a)2). 

III.  Theorem.  Every  covariant  of f  =  anr=bnv  =  •••  which  is 
obtainable  as  a  covariant  of  (/,  f)'u '  =  f/inrru  =  Qab)-ka",~-kb"--k 
(Chap.  II,  §  4)  is  congruent  to  any  definite  one  of  its  own  terms 
modulo  (ab)2k+i. 

The  form  of  such  a  concomitant  monomial  in  the  g  sym- 
bols is  A=  (g^Yig^y  ...gfsfc  «... 

Proceeding  by  the  method  of  Section  2  of  this  chapter  change 
g1  into  y  ;  i.e.  g\\  =  yv  9n=  —  Vv  Then  A  becomes  a  form  of 
order  2  n  —  4  k  in  y,  viz.  a2/1-4*  =  /3'-j'~ ik  =  •••.     Moreover 

A  =  (a2?-4*,  gi"y-u'?n-il'  =  «-**,  (a5)2X~2^lT2*)2w"U' 
by  the  standard  method  of  transvection.     Now  this  transvec- 
tant  A  is  free  from  y.     Hence  there  are  among  its  terms  ex- 
pressed in  the  symbols  of /only  two  types  of  adjacent  terms, 
viz.  (cf.  §  2,  II) 

(da)(eb)P,     (db)(ea)P. 


126  THE   THEORY   OF   INVARIANTS 

The  difference  between  A  and  one  of  its  terms  can  therefore 
be  arranged  as  a  succession  of  differences  of  adjacent  terms 
of  these  two  types  and  since  P  involves  (ab)2t  any  such  dif- 
ference is  congruent  to  zero  modulo  (a6)2i+1,  which  proves 
the  theorem. 

IV.   Theorem.      If  n  >4  k,  any  covariant  of  the  covariant 
c/2r"-u=  {abykan-2kb^-u 
is  expressible  in  the  form 

2Qhi+W'(*«?(«)^,  (152) 

where  C2k+X  represents  a  covariant  of  grade  2  k  +  1  at  least,  the 
second  term  being  absent  (T  =  0)  if  n  is  odd. 

Every  covariant  of  g2""11'  of  a  stated  degree  is  expressible 
as  a  linear  combination  of  transvectants  of  gf~ik  with  covari- 
ants  of  the  next  lower  degree  (cf.  §  2,  III).  Hence  the 
theorem  will  be  true  if  proved  for  T  =  (g2J'^k\  g2"~u'Y<  the 
covariants  of  second  degree  of  this  form.  By  the  fore- 
going theorem  ^is  congruent  to  any  one  of  its  own  terms 
mod  (aby2M.  Hence  if  we  prove  the  present  theorem  for  a 
stated  term  of  T,  the  conclusion  will  follow.  In  order  to 
select  a  term  from  T  we  first  find  T  by  the  standard  trans- 
vection  process  (cf.  Chap.  Ill,  §  2).  We  have  after  writing 
s  =  n—  2  k  for  brevity,  and  a^  =  arj 

a  fsV    s    \ 
T=  (aby*(cdy*X       }Z  ~  *'  c  £*#£*+*.  (ca)Xday-'a*-°.    (153) 


Now  the  terms  of  this  expression  involving  a  may  be  obtained 
by  polarizing  «;s  t  times  with  respect  to  y,  a  —  t  times  with 
respect  to  z.  and  changing  y  into  c  and  z  into  d.  Perform- 
ing these  operations  upon  a%b%  we  obtain  for  T, 


REDUCTION  127 

<x      I    cr — t 


t=0  «=0  »=0 

X  a™-vbs-'r+w+vc%-tds-'T+t,  (154) 

where  Ktuv  is  numerical.     Evidently  a  is  even. 

We  select  as  a  representative  term  the  one  for  which  t  =  a, 
u  =  v  =  0. 

This  is 

<j>  =  (aby^bcy(cdykanJr2kbnJr'2k-'Tcnlr  2*-*d»- 2k. 

Assume  n  ^  A.k.     Then  by  Section  6, 

yfr  =  ^abyk{bcy{caykanfikln-'2k-acnl-'2k-'T 

can   be    expressed   in    terms   of   covariants  whose  grade  is 

greater  than  2  k  unless  <r  =  2  k  =  - .     Also  in  the  latter  case 

-<|r  is  the  invariant 

yjr  =  (ab)Xbey(eay2. 

It  will  be  seen  at  once  that  n  must  then  be  divisible  by  4. 
Next  we  transform  </>  by  (cd~)aJ.  =  (ad')cx  —(ac)dx.  There- 
suit  is 

t=0  ^  ' 

(I)  Now  if  a>k,  we  have  from  Section  6  that  <£  is  of  grade 

n  n  n 

>  |  •  3  &,  i.e.  >  2&,  or  else  contains  (aJ)\6c)\ca)2,  i.e. 

0  =  2(72/6+1  +  (aby(bcyXcayr.  (155) 

(II)  Suppose  then  cr^k.  Then  in  <£',  since  i=  2  k  has 
been  treated  under  y\r  above,  we  have  either 

(a)  i  ^  &, 
or  (6)  2  k—  i>  k. 

In  case  (a)  (155)  follows  directly  from  Section  6.  In  case 
(by  the  same  conclusion  follows  from  the  argument  in  (I). 
Hence  the  theorem  is  proved. 


CHAPTER   V 

GORDANS  THEOREM 

We  are  now  in  position  to  prove  the  celebrated  theorem 
that  every  concomitant  of  a  binary  form /is  expressible  as  a 
rational  and  integral  algebraical  function  of  a  definite  finite 
set  of  the  concomitants  of  /.  Gordan  was  the  first  to  ac- 
complish the  proof  of  this  theorem  (1868),  and  for  this  rea- 
son it  has  been  called  Gordan's  theorem.  Unsuccessful 
attempts  to  prove  the  theorem  had  been  made  before  Gordan's 
proof  was  announced. 

The  sequence  of  introductory  lemmas,  which  are  proved 
below,  is  that  which  was  first  given  by  Gordan  in  his  third 
proof  (cf.  Vorlesungen  iiber  Invariantentheorie,  Vol.  2, 
part  3).*  The  proof  of  the  theorem  itself  is  somewhat 
simpler  than  the  original  proof.  This  simplification  has  been 
accomplished  bjT  the  theorems  resulting  from  Jordan's  lemma, 
given  in  the  preceding  chapter. 

SECTION  1.     PROOF  OF  THE  THEOREM 

We  proceed  to  the  proof  of  a  series  of  introductory  lemmas 
followed  by  the  finiteness  proof. 

I.  Lemma  1.  If  (A)  :  Av  Av  •••,  Ak  is  a  system  of  binary 
forms  of  respective  orders  av  aT  •••,  ak,  and  (B):  Bv  B2.  ■••, 
BP  a  system  of  respective  orders  hv  b2,  ■••,  bt.  and  if 

4>  =  A? A?  •  •  •  Ap,   ir  =  B% *B$*  -.-B?i 

*  Cf.  Grace  and  Young  ;  Algebra  of  Invariants  (1903). 
128 


GORDAN'S   THEOREM  129 

denote  any  two  products  for  which  the  as  and  the  fts  are  all 
positive  integers  (or  zero},  then  the  number  of  transvectants  of 
the  type  of 

which  do  not  contain  reducible  terms  is  finite. 

To  prove  this,  assume  that  any  term  of  t  contains  p  sym- 
bols of  the  forms  A  not  in  second  order  determinant  com- 
binations with  a  symbol  of  the  B  forms,  and  a  symbols  of  the 
_5's  not  in  combination  with  a  symbol  of  the  A's.  Then 
evidently  we  have  for  the  total  number  of  symbols  in  this 
term,  from  (A)  and  (B)  respectively, 

«1«1  +  <ha2  +   •"   +  akak  =  P+J, 

&1/91  +  -&a/8a+  •••  +6jA=o-  +  y. 

To  each  positive  integral  solution  of  the  equations  (156), 
considered  as  equations  in  the  quantities  a,  /3,  p,  <x,  /,  will 
correspond  definite  products  <£,  y]r  and  a  definite  index  j,  and 
hence  a  definite  transvectant  t.  But  as  was  proved  (Chap. 
IV,  §  3,  III),  if  the  solution  corresponding  to  (<£,  -^y  is  the 
sum  of  those  corresponding  to  (cf>v  -^1)?i  and  ($2,  -^2)S  then 
t  certainly  contains  reducible  terms.  In  other  words  trans- 
vectants corresponding  to  reducible  solutions  contain  re- 
ducible terms.  But  the  number  of  irreducible  solutions  of 
(15(3)  is  finite  (Chap.  IV,  §  5,  II).  Hence  the  number  of 
transvectants  of  the  type  t  which  do  not  contain  reducible 
terms  is  finite.  A  method  of  finding  the  irreducible  trans- 
vectants was  given  in  Section  3,  III  of  the  preceding 
chapter. 

Definitions.  A  system  of  forms  (J.)  is  said  to  be  com- 
plete when  any  expression  derived  by  convolution  from  a 
jiroduct  cf>  of  powers  of  the  forms  (^1)  is  itself  a  rational 
integral  function  of  the  forms  (^1). 

A  system  (A~)  will  be  called  relatively  complete  for  the 
modulus  G  consisting  of  the  product  of  a  number  of  sym- 
bolical determinants  when   any  expression  derived  by  con- 


130  THE    THEORY    OF    INVARIANTS 

volution  from  a  product  $  is  a  rational  integral  function  of 

the  forms  (J.)  together  with  terms  containing  Gr  as  a  factor. 

As  an  illustration  of  these  definitions  we  may  observe  that 

/=  a?  =  ...,  A  =  (_abyaxbx,   Q  =  {ab)\ca)bxc% 
R  =  (abf{ed)Xac~)(bd) 

is  a  complete  system.  For  it  is  the  fundamental  system  of 
a  cubic  /,  and  hence  any  expression  derived  by  convolution 
from  a  product  of  powers  of  these  four  concomitants  is  a 
rational  integral  function  of/,  A,  Q,  R. 

Again  /  itself  forms  a  system  relatively  complete  mod- 
ulo («6)2. 

Definition.  A  system  (A)  is  said  to  be  relatively  com- 
plete for  the  set  of  moduli  Grv  Cr2,  •••  when  any  expression 
derived  from  a  product  of  powers  of  A  forms  by  convolution 
is  a  rational  integral  function  of  A  forms  together  with 
terms  containing  at  least  one  of  the  moduli  Crv  GrT  •••  as  a 
factor. 

In  illustration  it  can  be  proved  (cf.  Chap.  IV,  §  7,  IV) 
that  in  the  complete  system  derived  for  the  quartic 

H=(abyalb% 

any  expression  derived  by  convolution  from  a  power  of  H 
is  rational  and  integral  in  H and 

ax  =  (ab)\    a2  =  (be)\ea)\ah  )2. 

Thus  His  a  system  which  is  relatively  complete  with  regard 
to  the  two  moduli 

ax  =  (ab)\   a2  =  (be}\eay(aby. 

Evidently  a  complete  system  is  also  relatively  complete 
for  any  set  of  moduli.     We  call  such  a  system  absolutely 

complete. 

Definitions.  The  system  ((7)  derived  by  transvection 
from  the  systems   (^4).   (i?)  contains  an  infinite  number  of 


GORDAN'S   THEOREM  131 

forms.  Nevertheless  (C)  is  called  a  finite  system  when  all 
its  members  are  expressible  as  rational  integral  algebraic 
functions  of  a  finite  number  of  them. 

The  system  (0)  is  called  relatively  finite  with  respect  to  a  set 
of  moduli  Grv  6r2,  •••  when  every  form  of  (C)  is  expressible 
as  a  rational  integral  algebraic  function  of  a  finite  number  of 
the  forms  (C)  together  with  terms  containing  at  least  one  of 
the  moduli  Grr  6r2,  •••  as  a  factor. 

The  system  of  all  concomitants  of  a  cubic  /  is  absolutely 
finite,  since  every  concomitant  is  expressible  rationally  and 
integrally  in  terms  of   /,  A,  Q,  R. 

II.  Lemma  2.  If  the  systems  (A),  (J5)  are  both  finite  and 
complete,  then  the  system  (C)  derived  from  them  by  transec- 
tion is  finite  and  complete. 

We  first  prove  that  the  system  (C)  is  finite.  Let  us  first 
arrange  the  transvectants 

t  =  ((f),  yfry 
in  an  ordered  array 

Tv   T2,    .-.,   Tf,    -..,  (157) 

the  process  of  ordering  being  defined  as  follows  : 

(a)  Transvectants   are    arranged    in    order    of   ascending 

total  degree   of  the  product  cf>y}r  in  the  coefficients  of  the 

forms  in  the  two  systems  (A),  (i?). 

(5)  Transvectants  for  which  the  total  degree  is  the  same 

are  arranged  in  order  of  ascending  indices  j ;    and  further 

than  this  the  order  is  immaterial. 

Now  let  t,  t'  be  any  two  terms  of  t.     Then 

where  $  is  a  form  derived  by  convolution  from  <£.  But  by 
hypothesis  (A),  (B)  are  complete  systems.  Hence  $,  i/r  are 
rational    and  integral  in  the  forms  A,  B  respectively, 

$=F(A),  f=  G(B). 


132  THE   THEORY   OF   INVARIANTS 

Therefore  ($,  \jr)J  can  be  expressed  in  terras  of  transvec- 
tants  of  the  type  r  {i.e.  belonging  to  ((7))  of  index  less 
than  j  and  hence  coming  before  t  in  the  ordered  array 
(157).  But  if  we  assume  that  the  forms  of  (C)  derived  from 
all  transvectants  before  t  can  be  expressed  rationally  and 
integrally  in  terms  of  a  finite  number  of  the  forms  of  ((7), 

then  all  C's  up  to  and  including  those  derived  from 

t  =  (<£,  yjry 
can  be  expressed  in  terms  of 

But  if  t  contains  a  reducible  term  t  =  txtv  then  since  tv  t2 
must  both  arise  from  transvectants  before  r  in  the  ordered 
array  no  term  t  need  be  added  and  all  C's  up  to  and  includ- 
ing those  derived  from  r  are  expressible  in  terms  of 

C    ('    ...    (y 

Thus  in  building  by  this  procedure  a  system  of  C's  in 
terms  of  which  all  forms  of  (C)  can  be  expressed  we  need  to 
add  a  new  member  only  when  we  come  to  a  transvectant  in 
(157)  which  contains  no  reducible  term.  But  the  number 
of  such  transvectants  in  (C)  is  finite.  Hence,  a  finite  num- 
ber of  C's  can  be  chosen  such  that  every  other  is  a  rational 
function  of  these. 

The  proof  that  ((7)  is  finite  is  now  finished,  but  we  may 
note  that  a  set  of  C's  in  terms  of  which  all  others  are  expres- 
sible may  be  chosen  in  various  ways,  since  t  in  the  above  is 
any  term  of  t.  Moreover  since  the  difference  between  any 
two  terms  of  t  is  expressible  in  terms  of  transvectants  be- 
fore t  in  the  ordered  array  we  may  choose  instead  of  a 
single  term  t  of  an  irreducible  r  =  ($,  y^y,  an  aggregate  of 
any  number  of  terms  or  even  the  whole  transvectant  and  it 
will  remain  true  that  every  form  of  (C)  can  be  expressed  as 


GORDAN'S   THEOREM  133 

a  rational  integral  algebraic  function  of  the  members  of  the 
finite  system  so  chosen. 

We  next  prove  that  the  finite  system  constructed  as  above 
is  complete. 

i^et  ^1'    ^-/21   *""'    ^r 

be  the  finite  system.  Then  we  are  to  prove  that  any  ex- 
pression X  derived  by  convolution  from 

is  a  rational  integral  algebraic  function  of  Ov  •  ••,  Cr.  Assume 
that  X  contains  p  second-order  determinant  factors  in  which 
a  symbol  from  an  (A)  form  is  in  combination  with  a  symbol 
belonging  to  a  (2?)  form. 

Then  X  is  a  term  of  a  transvectant  (</>,  -v/r)p,  where  (f>  con- 
tains symbols  from  system  (A)  only,  and  -v/r  contains  symbols 
from  (B)  only.  Then  <f>  must  be  derivable  by  convolution 
from  a  product  <f>  of  the  A's  and  yfr  from  a  product  i/r  of  B 
forms.     Moreover 

and  <£,  yjr  having  been  derived  by  convolution  from  <£,  ifr, 
respectively,  are  ultimately  so  derivable  from  cf>,  ty.     But 

4>  =  F(A)t  ^=  G(B), 

and  so  X  is  expressed  as  an  aggregate  of  transvectants  of  the 
type  of 

t  =  (</>,  \}r  )J. 

But  it  was  proved  above  that  every  term  of  t  is  a  rational 
integral  function  of 

Cj,  •••,  Cr. 

Hence  X  is  such  a  function ;   which  was  to  be  proved. 

III.  Lemma  3.  If  a  finite  system  of  forms  (-4),  all  the 
members  of  which  are  covariants  of  a  binary  form  f  includes  f 
and  is  relatively  complete  for  the  modidus  G' ;  and  if  in  addi- 
tion, a  finite  system  (B)  is  relatively  complete  for  the  modidus 


134  THE  THEORY   OF   INVARIANTS 

G  and  includes  one  form  Bx  whose  only  determinantal  factors 
are  those  constituting  G\  then  the  system  (C)  derived  by 
transvection  from  (A)  and  (B)  is  relatively  finite  and  complete 
for  the  modulus  G. 

In  order  to  illustrate  this  lemma  before  proving  it  let  (A) 
consist  of  one  form/  =  a§  =  •••,  and  (i?)  of  two  forms 

A  =(abyaxbx,  B  =  (ab)Xae')(bd)(cdy. 

Then  (A)  is  relatively  complete  for  the  modulus  G'  =  (aby. 
Also  B  is  absolutely  complete,  for  it  is  the  fundamental 
system  of  the  Hessian  of  f.  Hence  the  lemma  states  that 
((7)  should  be  absolutely  complete.  This  is  obvious.  For 
(C)  consists  of  the  fundamental  system  of  the  cubic, 

/,  A,   Q,  M, 

and  other  co variants  of/. 

We  divide  the  proof  of  the  lemma  into  two  parts. 

Part  1.  First,  we  prove  the  fact  that  if  P  be  an  expression 
derived  by  convolution  from  a  power  of  f  then  any  term,  t,  of 
o-  =  (P,  yfrY  can  be  expressed  as  an  aggregate  of  transvectants 
of  the  type 

t  =  (<f),  ^y, 

in  which  the  degree  of  <f>  is  at  most  equal  to  the  degree  of  P. 
Here  <f>  and  yfr  are  products  of  powers  of  forms  (A),  (B) 
respectively,  and  by  the  statement  of  the  lemma  (A)  con- 
tains only  covariants  of/ and  includes /itself. 

This  fact  is  evident  when  the  degree  of  P  is  zero.  To 
establish  an  inductive  proof  we  assume  it  true  when  the 
degree  of  P  is  <  r  and  note  that 

t  =  (P,  fy  +  2(P,  jrY  (%'  <  I), 

and,  inasmuch  as  P  and  P  are  derived  by  convolution  from 
a  J mwer  of/, 

P  =  F(A)  +  G'  Y=  F  (A~)  (mod  G'), 
P=  F'(A)  +  G'  Y'  =  F(A)  (mod  G'). 


GORDAN'S   THEOREM  135 

Also  ^  =  ®(B)  +  GZ  =  <P(B)  (mod  G). 

Hence  t  contains  terms  of  three  types  (a),  (ft),  (c). 

(a)  Transvectants  of  the  type  (F(A),  <E>(5))»,  the  degree 
of  -F(^4.)  being  r,  the  degree  of  P. 

(ft)  Transvectants  of  type  (Gr  Y,  f)\  G' Y  being  of  the 
same  degree  as  P. 

(e)  Terms  congruent  to  zero  modulo  G. 

Now  for  (a)  the  fact  to  be  proved  is  obvious.  For  (ft), 
we  note  that  G'  Y  can  be  derived  by  convolution  from  Bxf% 
where  s  <  r.  Hence  any  term  of  (6r'  Y,  yjr)k  can  be  derived 
by  convolution  from  B^-^r  and  is  expressible  in  the  form 

2(P',^), 

where  P'  is  derived  by  convolution  from  /*  and  is  of  degree 
<r.  But  by  hypothesis  every  term  in  these  latter  transvec- 
tants is  expressible  as  an  aggregate 

=  2(0,  fy  (modulo  #), 
inasmuch  as 

Btf=®(B)  (modulo  G). 

But  in  this  (<£,  -fry  </>  is  of  degree  ^  s  <  r.     Hence 

£=2(<£,  yjry  (mod  (7), 

and  the  desired  inductive  proof  is  established. 

As  a  corollary  to  the  fact  just  proved  we  note  that  if  P 
contain  the  factor  G',  then  any  term  in 

(^  +y 

can  be  expressed  in  the  form 

2(0,  f  y,  (158) 

where  the  degree  of  0  is  less  titan  that  of  P. 

Part  2.  We  now  present  the  second  part  of  the  proof  of 
the  original  lemma,  and  first  to  prove  that  (C)  is  relatively 
finite  modulo  G. 


136  THE   THEORY   OF   INVARIANTS 

We  postulate  that  the  transvectants  of  the  system  (C) 
are  arranged  in  an  ordered  array  defined  as  follows  by 
(a),  (5),  (C). 

(a)  The  transvectants  of  (C)  shall  be  arranged  in  order  of 
ascending  degree  of  (f>yfr,  assuming  the  transvectants  to  be  of 
the  type  t  =  (</>,  -»/r)>. 

(5)  Those  for  which  the  degree  of  <f>y]r  is  the  same  shall  be 
arranged  in  order  of  ascending  degree  of  $. 

(c)  Transvectants  for  which  both  degrees  are  the  same 
shall  be  arranged  in  order  of  ascending  index  j ;  and  further 
than  this  the  ordering  is  immaterial. 

Let  t,  t'  be  any  two  terms  of  t.     Then 

t'-t  =  ^^y  o''  <./)• 

Also  by  the  hypotheses  of  the  lemma 
4>  =  F(A)+G'Y, 

^r  =  <s>(B)+  az. 

Hence 

t'-t^^QFiA),  <t>(B)y  +  z(a>r,  <s>(Byy  (mod  a). 

Now  transvectants  of  the  type  (F(A~),  <£>(B)y  belong 
before  r  in  the  ordered  array  since  j'  <  j  and  the  degree  of 
F(A)  is  the  same  as  that  of  <f).  Again  (Gr'Y,  <$>(Byy'  can 
by  the  above  corollary  (158)  be  expressed  in  the  form 

where  the  degree  of  </>'  is  less  than  that  of  Gr'Y  and  hence 
less  than  that  of  (f>. 

Consequently  t'  —  t  can  be  written 

f  -t  =  2(4>",  ffy  +  2(#',  fy  (mod  ay 

where  the  degree  of  <f>"  is  the  same  as  that  of  <f>  and  where 
;'  <  j,  and  where  the  degree  of  <f>'  is  less  than  that  of  </>. 
Therefore  if  all  terms  of  transvectants  coming  before 

T  =  (<£.  yfry 


GORDAN'S   THEOREM  137 

in  the  ordered  array  are  expressible  rationally  and  integrally 

in  terms  of 

n     n  n 

Oj,     L/2,    •",     *-/g, 

except  for  terms  congruent  to  zero  modulo  Gr,  then  all  terms 
of  transvectants  up  to  and  including  r  can  be  so  expressed 
in  terms  of 

tV  C2,  •",  Cg,         t, 

where  t  is  any  term  of  t.  As  in  the  proof  of  lemma  2,  if  t 
contains  a  reducible  term  t=t-fiv  t  does  not  need  to  be 
added  to 

P     C     ...    O 

since  then  tv  t2  are  terms  of  transvectants  coming  before  r 
in  the  ordered  array.  Hence,  in  building  up  the  system  of 
Cs  in  terms  of  which  all  forms  of  ((7)  are  rationally  ex- 
pressible modulo  Gr,  by  proceeding  from  one  trans vectant  r 
to  the  next  in  the  array,  we  add  a  new  member  to  the  sys- 
tem only  when  we  come  to  a  transvectant  containing  no 
reducible  term.  But  the  number  of  such  irreducible  trans- 
vectants in  (0~)  is  finite.  Hence  (C)  is  relatively  finite 
modulo  Gr.  Note  that  Cv  •••,  Cq  may  be  chosen  by  select- 
ing one  term  from  each  irreducible  transvectant  in  (C). 

Finally  we  prove  that  (C)  is  relatively  complete  modulo 
6r.     Any  term  X  derived  by  convolution  from 

is  a  term  of  a  transvectant  (</>,  -i^)?,  where,  as  previously,  cj> 
is  derived  by  convolution  from  a  product  of  A  forms  and  yfr 
from  a  product  of  B  forms.     Then 

X  =  (£,  fy  +  2(5,  ^)p'  p'  <  p. 

That  is,  X  is  an  aggregate  of  transvectants  ($,  ^)CT,  <£  =  P 
can  be  derived  by  convolution  from  a  power  of/,  and 

-f  =  3>(i?)  (mod  G-). 


138  THE   THEORY   OF   INVARIANTS 

Tims,  X=  2  (P.  ®(B)Y  (mod  G) 

=  2( P,  ^)»  (mod  (7) 
=  £(<£,  -f );  (mod  6?) 

where  <£  is  of  degree  not  greater  than  the  degree  of  P,  by 
the  rirst  part  of  the  proof.  But  all  transvectants  of  the  last 
type  are  expressible  as  rational  integral  functions  of  a  finite 
number  of  C^'s  modulo  G.  Hence  the  system  (  C)  is  relatively 
complete,  as  well  as  finite,  modulo  G. 

Corollary  1.  If  the  system  (P)  is  absolutely  complete 
then  ((7)  is  absolutely  complete. 

Corollary  2.  If  (B)  is  relatively  complete  for  two 
moduli  Gv  6r2  and  contains  a  form  whose  only  determi- 
nantal  factors  are  those  constituting  (r',  then  the  system  (C) 
is  relatively  complete  for  the  two  moduli  Gr  G2. 

IV.  Theorem.  Tlie  system  of  all  concomitants  of  a  binary 
formf=  anx  =  •••  of  order  n  is  finite. 

The  proof  of  this  theorem  can  now  be  readily  accomplished 
in  view  of  the  theorems  in  Paragraphs  III,  IV  of  Chapter  IV, 
Section  7,  and  lemma  3  just  proved. 

The  system  consisting  of  /  itself  is  relatively  complete 
modulo  («5)2.  It  is  a  finite  system  also,  and  hence  it  satis- 
fies the  hypotheses  regarding  (^4)  in  lemma  3.  This  system 
(^4.)  =  / may  then  be  used  to  start  an  inductive  proof  con- 
cerning systems  satisfying  lemma  3.  That  is  we  assume 
that  we  know  a  finite  system  ^4fc_j  which  consists  entirely  of 
covariants  of  f  which  includes  f  and  which  is  relatively 
complete  modulo  (a5)2fc.  Since  every  covariant  of  /  can  be 
derived  from/ by  convolution  it  is  a  rational  integral  func- 
tion of  the  forms  in  Ak_l  except  for  terms  involving  the 
factor  (a6)2*.  We  then  seek  to  construct  a  subsidiary  finite 
system  Bk_x  which  includes  one  form  Bx  whose  only  deter- 
minant factors  are  (ab*)-k  =  G' ,  and  which  is  relatively  com- 
plete modulo   (abyk+2  =  G.     Then   the   system   derived  by 


GORDAN'S   THEOREM  139 

transvection  from  Ak_x  and  Bk_x  will  be  relatively  finite  and 
complete  modulo  (aby2k+2.  That  is,  it  will  be  the  system  Ak. 
This  procedure,  then,  will  establish  completely  an  inductive 
process  by  which  we  construct  the  system  concomitants  of/ 
relatively  finite  and  complete  modulo  (ab  )2k+2  from  the  set 
finite  and  complete  modulo  (a5)2fc,  and  since  the  maximum 
grade  is  n  we  obtain  by  a  finite  number  of  steps  an  abso- 
lutely finite  and  complete  system  of  concomitants  of/.  Thus 
the  finiteness  of  the  system  of  all  concomitants  of  /will  be 
proved. 

Now  in  view  of  the  theorems  quoted  above  the  subsidiary 
system  Bk_i  is  easily  constructed,  and  is  comparatively 
simple.     We  select  for  the  form  Bx  of  the  lemma 

Next  we  set  apart  for  separate  consideration  the  case  (<?) 
n  =  4  k.     The  remaining  cases  are  (a)  w>4  k,  and  (b)  n<  4  k. 

(«)  By  Theorem  IV  of  Section  7  in  the  preceding  chapter 
if  n>4&  any  form  derived  by  convolution  from  a  power  of 
hk  is  of  grade  2k +  1  at  least  and  hence  can  be  transformed 
so  as  to  be  of  grade  2  k +  2  (Chap.  IV,  §7,  II).  Hence  hk 
itself  forms  a  system  which  is  relatively  finite  and  complete 
modulo  (afoyk+2  and  is  the  system  Bk_x  required. 

(b)  If  n<4ik  then  hk  is  of  order  less  than  n.  But  in  the 
problem  of  constructing  fundamental  systems  we  may  pro- 
ceed from  the  forms  of  lower  degree  to  those  of  the  higher. 
Hence  we  may  assume  that  the  fundamental  system  of  any 
form  of  order  <  n.  is  known.  Hence  in  this  case  (b)  we 
know  the  fundamental  system  of  hk.  But  by  III  of  Chapter 
IV,  Section  7  any  concomitant  of  hk  is  congruent  to  any  one 
of  this  concomitant's  own  terms  modulo  (a6)2fc+1.  Hence  if 
we  select  one  term  from  each  member  of  the  known  funda- 
mental system  of  hk  we  have  a  system  which  is  relatively 
finite  and  complete  modulo  (ab)2k+2;  that  is,  the  required 
system  J5fc_1. 


140  THE   THEORY   OF   INVARIANTS 

(c)  Next  consider  the  case  n  =  4  k.  Here  b}r  Section  7.  IV 
of  the  preceding  chapter  the  system  B^  =  hk  is  relatively 
finite  and  complete  with  respect  to  two  moduli 

G1  =  (al>yk+\  G2  =  (ah)2k(beyk(eayk, 

and  G2  is  an  invariant  off.  Thus  by  corollary  2  of  lemma  3 
the  system,  as  Ck,  derived  by  transvection  from  Ak^1  and 
-Bjt-i  is  relatively  finite  and  complete  with  respect  to  the  two 
moduli  Gv  G2.  Hence,  if  Ck  represents  any  form  of  the 
system  Ck  obtained  from  a  form  of  Ck  by  convolution, 

Ok  =  Fl{Ck)+  G2PX  (mod (ab)^) . 

Here  P1  is  a  covariant  of  degree  less  than  the  degree  of  Ok. 
Hence  P1  may  be  derived  by  convolution  from/,  and  so 

Px  =  F2  (  Ck)  +  G2P2  (mod  (aby^y 

and  then  P2  is  a  covariant  of  degree  less  than  the  degree  <>f 
Pv  By  repetitions  of  this  process  we  finally  express  Ck  as  a 
polynomial  in 

G-2=(abyk(bcykyea)2'\ 

whose  coefficients  are  all  co variants  of  /  belonging  to  Ck, 
together  with  terms  containing  G1  =  {ab)2k+2  as  a  factor,  i.e. 

O.-F^CO  +  a2F2(Ck)  +  GlF3(Ck)  +  -  +  a2Pr(Ck) 

(mod  Gx). 

Hence  if  we  adjoin  G2  to  the  system  Ck  we  have  a  system 
Ak  which  is  relatively  finite  and  complete  modulo  (abyk+2. 

Therefore  in  all  cases  (a),  (&),  (e)  we  have  been  able  to 
construct  a  system  Ak  relatively  finite  and  complete  modulo 
(ai)2fc+2  from  the  system  Ak_l  relatively  finite  and  complete 
modulo  (abyk.  Since  A0  evidently  consists  of  /  itself  the 
required  induction  is  complete. 

Finally,  consider  what  the  circumstances  will  be  when  we 
come  to  the  end  of  the  sequence  of  moduli 

(aby.   (ab)\   (ab)G.  •••. 


GORDAN'S   THEOREM  141 

If  n  is  even,  n  =  2g,  the  system  A9_1  is  relatively  finite  and 
complete  modulo  (a&)2»  =  (ab)n.  The  system  Bg^  consists 
of  the  invariant  (ab)n  and  hence  is  absolutely  finite  and 
complete.  Hence,  since  Ag  is  absolutely  finite  and  complete, 
the  irreducible  transvectants  of  Ag  constitute  the  funda- 
mental system  of  /.  Moreover  Ag  consists  of  Ag_1  and  the 
invariant  (ab')n. 

If  n  is  odd,  n=  2g  -f-  1,  then  Ag_x  contains  /  and  is  rela- 
tively finite  and  complete  modulo  (ab}2a.  The  system  Bg_A 
is  here  the  fundamental  system  of  the  quadratic  {abyl9axbx 
e.g. 

Bg^  =  {(ab)2°axbx,  (aby^(ac)(bd)(edya\. 

This  system  is  relatively  finite  and  complete  modulo  (ab)2o+1. 
But  this  modulus  is  zero  since  the  symbols  are  equivalent. 
Hence  Bg_x  is  absolutely  finite  and  complete  and  by  lemma 
3  Ag  will  be  absolutely  finite  and  complete.  Then  the  set 
of  irreducible  transvectants  in  Ag  is  the  fundamental  system 
of/. 

Grordan's  theorem  has  now  been  proved. 

SECTION   2.     FUNDAMENTAL   SYSTEMS   OF   THE   CUBIC 
AND  QUARTIC   BY   THE   GORDAN   PROCESS 

It  will  now  be  clear  that  the  proof  in  the  preceding  section 
not  only  establishes  the  existence  of  a  finite  fundamental 
system  of  concomitants  of  a  binary  form  /  of  order  n,  but  it 
also  provides  an  inductive  procedure  by  which  this  system 
may  be  constructed. 

I.  System  of  the  cubic.   For  illustration  let  n  =  3, 
/=4  =  53=.... 

The  system  A0  is/  itself.  The  system  i?0  is  the  fundamental 
system  of  the  single  form 

ftj  =  (abyaj)^ 


142  THE   THEORY   OF   INVARIANTS 

since  hl  is  of  order  less  than  3.     That  is, 

B0  =  \(abfaxbx,  D\ 

where  D  is  the  discriminant  of  hr  Then  Ax  is  the  system 
of  transvectants  of  the  type  of 

T  =  {f\htl)yy. 

But  B0  is  absolutely  finite  and  complete.  Hence  Ax  is  also. 
Now  D  belongs  to  this  system,  being  given  by  «=  /3  =  j 
=  0,  7  =  1.  If  j  >  0  then  t  is  reducible  unless  7=0,  since 
D  is  an  invariant.  Hence,  we  have  to  consider  which  trans- 
vectants 

r  =  (/a,  h\y 

are  irreducible.  But  in  Chapter  IV,  Section  3  II,  we  have 
proved  that  the  only  one  of  these  transvectants  which  is  ir- 
reducible is  Q  =  (/,  /jj).  Hence,  the  irreducible  members 
of  Ax  consist  of 

A,  =  {/,  hv  Q,  D\, 

or  in  the  notation  previously  introduced, 

A1=\f,  A,  Q,  R\. 

But  B0  is  absolutely  complete  and  finite.  Hence  these 
irreducible  forms  of  Ax  constitute  the  fundamental  system 
of/. 

II.    System  of   the  quartic.     Let/=  a|=  b%  —  •••.     Then 
A0  =  \f\.     Here  B0  is  the  single  form 

\  =  (ab)2axbx, 

and  B0  is  relatively  finite  and  complete  (mod4  («#)* 
( <ib)2(bcy(ca~)2).     The  system  Gx  of  transvectants 

is  relatively  finite  and  complete  (modd  (a J)4,  (rt5)2(fo)2(cfl02). 
In  t  if  _/>  1,  t  contains  a  term  with  the  factor  (afi)2(rtc)2 
which  is  congruent  to  zero  with  respect  to  the  two  moduli. 


GORDAN'S   THEOREM  143 

Hence  j  —  1,  and  by  the  theory  of  reducible  transvectants 
(Chap.  IV,  §  3,  III) 

4a  —  4  <  /  <  4  a, 

or  a  =  1,  ft  =  1.  The  members  of  Cx  which  are  irreducible 
with  respect  to  the  two  moduli  are  therefore 

/,  K  (/,  *,). 

Then  ^  =  \f,  hv  (/,  7^),  <7=  (a6)2(5c)2(^)2l- 

Next  5t  consists  of  i  =  (a6)4  and  is  absolutely  complete. 
Hence,  writing  hx  =  II,  (/,  Ax)=  I7,  the  fundamental  system 
of/ is 

/,  H,  T,  i,  J. 


CHAPTER   VI 

FUNDAMENTAL  SYSTEMS 

In  this  chapter  we  shall  develop,  by  the  methods  and  pro- 
cesses of  preceding  chapters,  typical  fundamental  s}Tstems 
of  concomitants  of  single  forms  and  of  sets  of  forms. 

SECTION    1.     SIMULTANEOUS   SYSTEMS 

In  Chapter  V,  Section  1,  II,  it  has  been  proved  that  if  a 
system  of  forms  (A)  is  both  finite  and  complete,  and  a  sec- 
ond system  (i?)  is  also  both  finite  and  complete,  then  the 
system  ($)  derived  from  (A)  and  (i?)  by  transvection  is 
finite  and  complete.  In  view  of  Gordan's  theorem  this 
proves  that  the  simultaneous  system  of  any  two  binary 
qualities/,  g  is  finite,  and  that  this  simultaneous  system  may 
be  found  from  the  respective  systems  of  /  and  g  by  trans- 
vection.    Similarly  for  a  set  of  n  qualities. 

I.  Linear  form  and  quadratic.  The  complete  system  of 
two  linear  forms  consists  of  the  two  forms  themselves  and 
their  eliminant.     For  a  linear  form  I  =  lr,  and  a  quadratic 

/,  we  have 

(4)=  Z,  (B)=\t\D\. 

Then  S  consists  of  the  transvectants 

S=  \(faD?,  lyyi. 

Since  D  is  an  invariant  S  is  reducible  unless  /3  =  0.  Also 
8<7,  and  unless  8  =  7,  (/",  lyY  is  reducible  by  means  of  the 
product 

Hence  7  =  8.     Again,  by 

144 


FUNDAMENTAL   SYSTEMS  145 

S  is  reducible  if  S>2.      Hence  the  fundamental  system  of  / 
and  Z  is 

S=\f,D,l,  (/,  Z),(/,  z2)2!- 

When  expressed  in  terms  of   the    actual    coefficients    these 
forms  are 

l  =  d(yl\  +  &-{£<L  —   'x  =  \r  =    " '  1 

f=  lp\  +  2  Vi^  +  hxl  =  «l  =  hl  =  "• 
D=2(b0b2-bl)=(ab)\ 

(/>   0  =  (Vl  -  Vo>l  +  ( Vl  -  Vo>2=  OO®*' 

(/,  P)2  =  bQa\  —  2  b1a0a1  4-  Z>2aj)  =  (al)(aV). 

II.  Linear  form  and  cubic.     If  l=lx  and  /=  a%  =  J|  =  •••, 
then  (cf.  Table  I), 

and  S  =  (faA^QyJl%lsy. 

Since  i2  is  an  invariant  e  =  0  for  an  irreducible  transvectant. 
Also  7]  =  8  as  in  (I).     If  a  =#=  0  then,  by  the  product 

(/,  pyHr-WQ^  i*-*y-\ 

S  is  reducible  unless  B  <  3,  and  if  S  <  3  #  is  reducible  by 
(/.W-'A^,  1)°; 

unless  /3  =  7  =  0,  a  =  1.     Thus  the  fundamental  system  of  / 
and  Z  is 

S  =  \f,  A,  ft  £,  Z,  (/,  0,  (/,  Z2)2,  (/,  Z3)3, 
(A,  Z),  (A,  Z2)2,  (ft  0,  (ft  Z2)2,  (ft  Z3)3;. 

III.  Two   quadratics.     Let  /=  a%  =  <2 ;  g  =  Z>2  =  6'2  =  .... 
Then 

(4)  =  [/,  Djj,  (J5)  =  {<fc  D2|,  #  =  (/°2>f,  ^2>|)e. 
Here /3=  8=0.     Also 

2«^e^2«-l, 
27=e>27-l, 


140  THE   THEORY   OF   INVARIANTS 

and  consistent  with  these  we  have  the  fundamental  system 

S=\f,g,DvDvu\g),(f,g)2\. 

Written  explicitly,  these  quantities  are 
f  =  a0x'j  +  2  a^x^x^  +  a\x\  =  a  J  =  a'}  =  •  • ., 

9  =  h/i  +  -  hxix2  +  hxl  =  &$  =  U?  =  •••, 
2>1  =  2(a0a2-«f)  =  (aa')2, 

J=(f,9) 

h  =  (/,  </)2  =  «o^2  ~  2  «A  +  (hh  =  (ab)2- 

IV.   Quadratic  and  cubic.     Consider  next  the  simultaneous 
system  of  /=  ay.  =  a'}  =  •  ••,  g  =  b%  =  1<J  =  •  ••.     In  this  case 

(A)=  {/,  D\,  (B)=  \g,  A,  Q,  R\,  S  =  {faD\  g°A<>Q°R*y. 

In  order  that  S  may  be  irreducible,  ft  =  d  =  0.  Then  in 
case  7>2  and  £=£0,  S  =  (f%  #aA6Qc)Y  is  reducible  by  means 
of  the  product 

Hence  only  three  types  of  transvectants  can  be  irreducible  ; 

(/,A),  (/,A)2,  (f%g°Q*y. 

The  first  two  are,  in  fact  irreducible.  Also  in  the  third 
type  if  we  take  c  =  0,  the  irreducible  transvectants  given  by 
(/",  gay  will  be  those  determined  in  Chapter  IV,  Section 
3,  III,  and  are 

/,  g,  (/,  g),  (/,  g)\  (/*,  gy,  (/S,  ,,2)6. 

If  <?>1,  we  may  substitute  in  our  transvectant  (/*,  gaQcy 

the  svzygv 

02=-i(A3  +  %2); 

and  hence  all  transvectants  with  c  >  1  are  reducible.  Tak- 
ing a  =  0,  e  =  1  we  note  that  (/,  Q)  is  reducible  because  it 


FUNDAMENTAL   SYSTEMS 


147 


is  the  Jacobian  of  a  Jacobian.  Then  the  only  irreducible 
cases  are 

(/,  <?)*»  C/"i  Q)s> 

Finally  if  c  =  1,  a=f=  0,  the  only  irreducible  transvectant  is 

Therefore  the  fundamental  system  of  a  binary  cubic  and  a 
binary  quadratic  consists  of  the  fifteen  concomitants  given 
in  Table  III  below. 

TABLE    III 


Deurf.e 

Order 

0 

] 

2 

3 

1 

/ 

0 

2 

I) 

(/,  (j)2 

A 

(/,  g) 

3 

(/,  A)- 

(/'2,  </)3 

(/,  A) 

Q 

4 

2? 

(/,  e)2 

5 

(/8,  <72)6 

(.r2,  e)8 

7 

(A  </<2)6 

SECTION   2.     SYSTEM   OF   THE   QUINTIC 

The  most  powerful  process  known  for  the  discovery  of  a 
fundamental  system  of  a  single  binary  form  is  the  process  of 
Gordan  developed  in  the  preceding  chapter.  In  order  to 
summarize  briefly  the  essential  steps  in  this  process  let  the 
form  be/.  Construct,  then,  the  system  A0  which  is  finite 
and  complete  modulo  (a?>)2,  i.e.  a  system  of  forms  which  are 
not  expressible  in  terms  of  forms  congruent  to  zero  modulo 
(a5)2.  Next  construct  Av  the  corresponding  system  modulo 
(aJ)4,  and  continue  this  step  by  step  process  until  the  system 
which  is  finite  and  complete  modulo  (ab)n  is  reached.  In 
order  to  construct  the  system  Ak  which  is  complete  modulo 
(a6)2fc+2  from  Ak_v   complete    modulo    (a5)2fc,  a    subsidiary 


148  THE   THEORY   OF   INVARIANTS 

system  Bk_1  is  introduced.  The  system  Bk_l  consists  of 
covariants  of  cf>  =  (ab)2ha"~2kb"~2k.  If  2  n  —  4  k  <  n  then  Bk_x 
consists  of  the  fundamental  system  of  <f>.  If  2  n  —  4  k  >  h, 
-Bi—!  consists  of  $  itself,  and  if  2  w  —  4  k  =  w,  .B^  consists 

w  n  n 

of  $  and  the  invariant  (a6)2(fo?)2(ra)2.  The  system  derived 
from  -Afc.j,  -Bi_!  by  trans vection  is  the  system  Ak. 

I.  The  quintic.  Suppose  that  w  =  5  ;  /=  a|  =  6|=  .... 
Here,  the  system  A0  is  /  itself.  The  system  B0  consists  of 
the  one  form  H =  («5)2a|J|.  Hence  the  system  Ax  is  the 
transvectant  system  given  by 

By  the  standard  method  of  transvection,  if  y  >  2  this  trans- 
vectant always  contains  a  term  of  grade  3  and  hence,  by  the 
theorem  in  Chapter  IV,  it  may  be  transformed  so  that  it 
contains  a  series  of  terms  congruent  to  zero  modulo  (aft)4, 
and  so  it  contains  reducible  terms  with  respect  to  this  modu- 
lus. Moreover  (/,  J3")2  is  reducible  for  forms  of  all  orders  as 
was  proved  by  Gordan's  series  in  Section  1  of  Chapter  IV. 
Thus  A1  consists  of/,  K  (f,  IT)  =  T. 

Proceeding  to  construct  Bx  we  note  that  i  =  (ab~)^axbx  is  of 
order  <  5.     Hence  Bx  consists  of  its  fundamental  system  : 

Bx=  \i,D], 

where  B  is  the  discriminant  of  i.  Hence  A2  which  is  here 
the  fundamental  system  of  /  is  the  transvectant  system 
given  by 

The  values  a  =  /3  =  y  =  8  =  T]=Q,  e  =  1  give  B.  Since  B 
is  an  invariant  </>  is  reducible  if  i)  =f=  0  and  e  =?=  0.  Hence 
e=  0. 

If  /3  >  1,  0  is  reducible  by  means  of  such  products  as 

(faHTy,  i)(Hfi-\  i5"1)"-1. 


FUNDAMENTAL   SYSTEMS  149 

Hence 

(i)  £=0 
.  (ii)   a  =  0,  7  =  0,/3  =  l. 

By  Chapter  IV,  Section  4,  IV, 

r2  =  -  \\<if,fyH*-Kf,  HyfH+(H,  nyp\. 

Hence 

T2  =  -  iff3  (mod  (aby). 

But  if  7  >  1,  the  substitution  of  this  in  </>  raises  j3  above  1 
and  hence  gives  a  reducible  transvectant.  Thus  7  =  0  or 
1  (cf.  Chap.  V  (158)). 

Thus    we    need   to   consider  in   detail  the  following  sets 
only  : 

(i)  a  =  1  or  2,  /3  =  0,  7  =  0, 
(ii)  a  =  0,  /3  =  0,  7  =  1, 
(iii)  «  =  1,  /3=0,  7  =  1, 
(iv)  «=0,/3=l,7  =  0. 

In  (i)  we  are  concerned  with  (/%  z5)v.     By  the  method  of 
Section  3,  Chapter  IV, 

2S-1^7<2S, 
5a  —  4  <  7  <  5  «, 

and  consistent  with  this  pair  of  relations  we  have 

<./,  </i  0.  (/.  02,  (/,  *'2)3.  c/.  vy,  (/.  *3)5. 

CA  *'3)6,  (/2,  *'4)7>  CA  *'4)8>  C/2> *'5)9^  C/2.  *"5)10- 

Of  these,  (/2,  i3)6  contains  reducible  terms  from  the  product 

a  *'2)4(/,  02, 

and  in  similar  fashion  all  these  transvectants  are  reducible 
except  the  following  eight: 

/,  i   (/,  .'),   (/,  02,   (/,  .1)8    (/,  ."2)4,   (/,  .'3)5,   (/2,  ,'5)10. 

In  (ii)  we  have   (2V8).     But  I7^  -  (aby(bc)axb*4,  and 
(T,  z)  contains  the  term  £=  —  (aby(bc}(bi)axbxcxix.     Again 

(6<0(k>A  =  H(^)2*'2  +  (KM  -  002&i]- 


150 


THE   THEORY   OF   INVARIANTS 


Hence  t  involves  a  term  having  the  factor/.  The  analysis 
of  the  remaining  cases  proceeds  in  precisely  the  same  way  as 
in  Cases  (i),  (ii).  In  Case  (ii)  the  irreducible  transvec- 
tants  prove  to  be 

(T,  i~)\  (r,  t*)*  (Z  *'3)6,  CA  i*)8,  (T,  f°)»- 

Case  (iii)  gives  but  one  irreducible  case,  viz.  (fT,  i7)14. 
In  Case  (iv)  we  have 

Off,  o,  (ff,  o2,  (#,  i^y,  (js;  ?y,  (H,  py,  (K  py. 

Table  IV  contains  the  complete  summary.  The  fundamen- 
tal system  of/ consists  of  the  23  forms  given  in  this  table. 

TABLE   IV 


De- 
gree 

Ordeb 

« 

0 

1 

2 

3 

4 

5 

6 

7 

9 

1 

/ 

2 

i 

JT 

3 

0\/)2 

(<,/) 

T 

i 

D 

(i,  H)* 

(i,  J?) 

5 

(*2,/)4 

(*2,/)3 

(/.  r)« 

6 

(i2,  H)* 

{i^Hf 

7 

(*'3,/)5 

(*2,  r)4 

s 

(i3,  H)6 

(i3,  i/)5 

9 

(**,  r)« 

11 

(i*,  r)8 

12 

(i6,/2)10 

13 

(#,  T)> 

18 

(I?, /T) " 

FUNDAMENTAL    SYSTEMS  151 

SECTION   3.     RESULTANTS   IN   ARONHOLD'S   SYMBOLS 

In  order  to  express  the  concomitants  derived  in  the  preced- 
ing section  in  symbolical  form  the  standard  method  of 
transvection  may  be  employed  and  gives  readily  any  con- 
comitant of  that  section  in  explicit  symbolical  form.  We 
leave  details  of  this  kind  to  be  carried  out  by  the  reader. 
However,  in  this  section  we  give  a  derivation,  due  to  Clebsch, 
which  gives  the  symbolical  representation  of  the  resultant  of 
two  given  forms.  In  view  of  the  importance  of  resultants  in 
invariant  theories,  this  derivation  is  of  fundamental  conse- 
quence. 

I.  Resultant  of  a  linear  form  and  an  n-ic.  The  resultant  of 
two  binary  forms  equated  to  zero  is  a  necessary  and  sufficient 
condition  for  a  common  factor. 

Let  f=  ax,  <f>  =  ax  =  axx^  +  a2x2  =  0. 

Then  xx :  x2  =  —  a2  :  ar     Substitution  in  /  evidently  gives  the 
resultant,  and  in  the  form 

R  =  (aa)n. 

II.  Resultant  of  a  quadratic  and  an  n-ic.     Let 

0  =  a|  =pxqx. 

The  resultant  R  =  0  is  evidently  the  condition  that  /  have 
either  px  or  qx  as  a  factor.     Hence,  by  I, 

R  =  (ap)n(bq)n. 

Let  us  express  R  entirely  in  terms  of  a,  5,  •••,  and  cc,  /3.  ■•• 
symbols. 

We  have,  since  a,  b  are  equivalent  symbols, 

R  =  i  { (ap~)n(bq)n  +  (aq)n(bpy  j . 

Let   (ap}(bq)=  fi,   (aq)(bp~)  =  v,  so  that 


152  THE   THEORY   OF   INVARIANTS 

Iheokem.  If  n  is  even,  It  =  £_Z —  is  rationally  and  inte- 
grally expressible  in  terms  of  p2  =  (p.—  v)2  and  a  =  p,v.  If 
n  is  odd.  (jjb  +  v)~lR  is  so  expressible. 

In  proof  write 

Sk=  fj.k+  (— 1)»-V. 

Then  B  =  iS„. 

Moreover  it  follows  directly  that 

&n  =    O  -  v  )^n-i  +   V-vSn-o? 


Also  for  w  even 

S\  =f*  —  v,  *S'0  =  2, 
and  for  n  odd 

>S\  =  /*  +  »,  #0  =  0. 
Now  let 

=  ptfj  +  cr,S'0  +  zp,S'2  +  20-^  +  z2pS3  +  22(7,S'2  + .... 
Then  we  have 

ft  =  P(  s\  +  zn)  +  cr(>v0  +  z&\  +  z2n ). 

and  ^  _  Q  +  o-g)!^,  +  aS0  _ 

1  —  pz  —  erg2 

Then  #„  is  the  coefficient  of  zn~2  in  the  expansion  of  ft. 

Now 

1  1        ]         az2         |         g¥ 

1  —  /3Z  —  <TZ2       1—  pz        (1  —  /32l)2        (1  —  pz)3 

=  l+pZ  +  p2z2  +  ph*+  ... 
+  (U2/)2+.3  ^2+  4  /r%3  +  •••  )<TZ2 

+ ( x + hi pz + hi p2z2 + hi p¥>2*4 
+ 

=  K0  +  Kxz  +  K2z2  +  Kf  +  -  , 


FUNDAMENTAL    SYSTEMS  153 

where 

if0=l,    K2  =  p*  +  <r,    Ki  =  p*+3p*cr+^ 

Kh  =  ph  +  (A  -  1)^-2  +  ^-2)(A-3)  ff2pA_4 

g-3)(/*-4)(A-5)  S6 

1.2-3  p  ' 

But 

Cl=\(pS1  +  *8Q)+z<rS1\\KQ  +  K1z  +  K2z*  +  ...  f. 

In  this,  taking  the  coefficient  of  zn~2, 

2E  =  Sn=  (p&\  +  aS0)Kn_2  +  o-^^n_  3. 
But, 

Hence, 

i£  =  l\S\Kn_1  +  o-*S'0ifn_2f . 

Hence  according  as  n  is  even  or  odd  we  have 

O    7?  n     ,  n-2    i    ^(n— 3)     o    „_4    ,    uCu—  4)(ft  —  5)      o    n_fi    , 

2  R  =  pn  +  ncrpn  2+^ — fr1*"?    4  +  ^— : — ^^ L^pn  b  +  ---, 

2  72=  O  +  i/)  \pn~l  +  («  -  2)crp»-3  +  <>-3)(>- 4)^-5 


Q  -4)(w-5)(w-6)   3      7  , 

1-2.3  P       +      U 


which  was  to  be  proved. 
Now  if  we  write 

<f>=Pxqx=<x%  =  /3$= •••» 

we  have 

Pich  =  av  Pi%+Ptfi  =  2  aiav  P&i  =  al- 
Then 

|t*+v=  (ap)(l>q~)  +  (aq)(bp) 

=  (a^2  -  «2PiX5i?2  -  %i)  +  C^i^  -  a2<2i)(t>iP2  -  hPi) 
=  2[a151«|  —  rt1?>2«1«2  —  rt2^1«1«2  +  «2^20Cf] 
=  2(aa)(5a), 


154  THE   THEORY   OF   INVARIANTS 

fiv  =  <r  =  (ap)(aq)(bp)(bq) 
=  paqa  •  2h9b  =  («a)2(«^)2, 
O  -  „)*  =p2=  \{ap-)(bq)  -  {aqXbp)\2=  (at>)\pq)2 
=  -  2(a6)2(  «/3)2  =  -  2(a?>  )2D. 

Let  the  symbols  of  0  be  a',  a"  •  ••;  /3',  /3",  ..-,7,  ....  Then 
we  can  write  for  the  general  term  of  R< 

pn-uak  _  (fJL  _  vy-u(fJLv)k=  (  _  Zf*  Dr\aby-2* 

x  (au'y(b/3')%aa")2(b/3"y  •••  (aa'*1)2^*")2 

=  (-2)2   *&  kAk. 

Evidently  Ak  is  itself  an  invariant.  When  we  substitute  this 
in  2  R  above  we  write  the  term  for  which  k  =  ^  n  last-  This 
term  factors.     For  if 

5  =  (aa')2(««")2-(a«     )2 

=  (6/8')2W')2-C^V, 

n 

then  o-5  =  J92. 

Thus  when  n  is  even, 

n  n — 2  n — 2    n — 4 

22=(-2)y.2^^+<-2>)'~2~2^11 

M(> -4)(M-o)  ,      m— .-,—  . 
+  "      1.2.3     ~C        ^      " 


(159) 


We  have  also, 

p*-2*-V*(>+i/)  =  2(-2)  2    "i>  -     "A-, 

where  ^.fc  is  the  invariant, 

4*  =  (a6)"-1-2X(a7)(i7)  •  (aa')2(W')2  •••  (aa<*>)2 


FUNDAMENTAL    SYSTEMS  155 

In  this  case, 

R=(-2D)~A,+  (n-2)(-2Dp~A1 

+  (n-3)(n-4)(_2J>)^     (159i) 

+  ... -*=1  DA^+A^. 

Thus  we  have  the  following : 

Theorem.  TJie  resultant  of  a  form  of  second  order  with 
another  form  of  even  order  is  ahvays  reducible  in  terms  of  in- 
variants of  loiver  degree,  but  in  the  case  of  a  form  of  odd  order 
this  is  not  proved  owing  to  the  presence  of  the  term  An_v 

A  few  special  cases  of  such  resultants  will  now  be  given ; 
(a),  (6),  Or),  (d). 
(a)  n  =  1 :  R  =  A0,  A0  =  (aa)2. 
(&)  n  =  2:R=-DA0  +  B\  A0=(aby,  B  =  (aay. 
R  =  -  (a/3)2(a6)2+(aa)2(W2- 

(c)  n=3:R=-2DA0  +  Av  A0  =  (aby(ay)(by). 

i1=(a7)({7)(a«)W2 
R  =  -  2(a^)\aby(ay)  (by)  +  (ay)  (by)  (aa)\b^f. 

(d)  n=±:R=2I)2A0-4DA1  +  B2,  A0  =  (aby. 

A1  =  (aby(aa)\b/3y. 
B  =  (aay(aa'y. 
R  =  2 (a/S)\a'/3f  y(aby-  4 («/3)2<>6)2<W )  W )2 
+  (a«)2(a«')2(^)2(5/3')2. 

SECTION  4.     FUNDAMENTAL   SYSTEMS   FOR   SPECIAL 
GROUPS  OF   TRANSFORMATIONS 

In  the  last  section  of  Chapter  I  we  have  called  attention 
to  the  fact  that  if  the  group  of  transformations  to  which  a 
form  /  is  subjected  is  the  special  group  given  by  the  trans- 
formations 


156  THE   THEORY   OF   INVARIANTS 

_  sin(ft)  —  a)  j      sin(&>  —  /3)    ,_        _  sin  «     .      sin /3    , 
sin  &)  sin  &)  "      sm  w         sin  &)    " 

then 

<^  =  .r2  -f  2  ajj^cos  &)  +  ./•:> 

is  a  universal  covariant.  Boole  was  the  first  to  discover 
that  a  simultaneous  concomitant  of  q  and  any  second  binary 
quantic  /  is,  when  regarded  as  a  function  of  the  coefficients 
and  variables  of  /,  a  concomitant  of  the  latter  form  alone 
under  the  special  group.  Indeed  the  fundamental  simulta- 
neous system  of  q  and  /  taken  in  the  ordinary  way  is,  from 
the  other  point  of  view,  evidently  a  fundamental  system  of/ 
under  the  special  group.  Such  a  system  is  Qalled  a  Boolean 
system  of/.  We  proceed  to  give  illustrations  of  this  type 
of  fundamental  system. 

I.  Boolean  system  of  a  linear  form.  The  Boolean  system 
for  a  linear  form, 

is  obtained  by  particularizing  the  coefficients  of/  in  Paragraph 
I,  Section  1  above  by  the  substitution 

%     h  ,  V 

,1,     COS  ft),      1 

Thus  this  fundamental  system  is 

q  =  x\  +  2  xxx2  cos  ft)  +  a*2,, 

a  =  sin2  ft), 

b  =  (a0  cos  ft)  —  a1)z1  +  (a0  —  ax  cos  &))x2, 

e  =  a2,  —  2  a^j  cos  &>  +  a\. 

II.  Boolean  system  of  a  quadratic.  In  order  to  obtain  the 
corresponding  system  for  a  quadratic  form  we  make  the 
above  particularization  of  the  b  coefficients  in  the  simulta- 
neous system  of  two  quadratics  (cf.  Section  1,  III  above). 


FUNDAMENTAL   SYSTEMS  157 

Thus  we  find  that  the  Boolean  system  of/ is 
/  =  aQx\  +  2  rtjZjZg  4-  a2x%, 
q  =  x\  +  2  a^g  cos  so  +  a;|, 
D  =  2(a0a2-  af), 
6?  =  sin2  &), 

e  =  a0  +  a2  —  2  Oj  cos  &>, 
#  =  (a0  cos  to  —  a^ajf  +  («0  —  a^)x^c%  +  (ax—  a2  cos  o  )x\ 

III.  Formal  modular  system  of  a  linear  form.  If  the 
group  of  transformations  is  the  finite  group  formed  by  all 
transformations  Tp  whose  coefficients  are  the  positive  residues 
of  a  prime  number  p  then,  as  was  mentioned  in  Chapter  I, 

J-J    ■/■■  i  ■'  O  \  *^  9 

is  a  universal  covariant.  Also  one  can  prove  that  all  other 
universal  covariants  of  the  group  are  covariants  of  L. 
Hence  the  simultaneous  system  of  a  linear  form  I  and  i, 
taken  in  the  algebraic  sense  as  the  simultaneous  system  of  a 
linear  form  and  a  form  of  order  p  +  1  will  give  formal 
modular  invariant  formations  of  I.  We  derive  below  a  fun- 
damental system  of  such  concomitants  for  the  case  p  =  3. 
Note  that  some  forms  of  the  system  are  obtained  by 
polarization.  Let/=  a0x1  +  «^2  ;  p  =  3.  The  algebraical 
system  of  /  is  /itself.      Polarizing  this, 


D  =  (  a3 —  J  /  =  ahc,  +  cvkx* 


The  fundamental  system  of  universal  covariants  of  the  group 

(is 

L=  x\x%  —  xxx%,    Q  =  x\-\-  x\x\  +  x\x\  +  x\  =  ((L,  X)2,  L). 


7*3  is 


160  THE   THEORY   OF   INVARIANTS 

If  we  raise  both  sides  of  this  identity  to  the  with  power 
we  have  at  once  the  symbolical  representation  of  the  typical 
representation  of/',  in  the  form 

where 

B0  =  (rt/f)m,  B1=(afM)m-l(a\),  B2  =  (a/0»-2(aX)a,  •  ••, 
Bm  =  (a\)'\ 
Also 

(\/x)m  =  Jm. 

Now  with  fi  =  (.ry )  we  have 

by  Euler's  theorem.     Moreover  we  now  have 

B0  =  a";.  =/,  7^  =  of-^oX),  B2  =  af-Xa\y,  •••, 
for  the  associated  forms,  and 

Pi 
and 

<j>(a0,  av  •  ••  ;  xv  z2)  =  —  </>(/,  -  2?r  Bv  •  •• ;  </>r  0). 
Pi 

Again  a  further  simplification  may  be  had  by  taking  for 
</>!  the  form /itself.     Then  we  have 

BQ  =f,  Bl=(ab)a»rlbnrl  =  0.  B2  =  (ab}(ac)a^>-2b'r1c>r\  ••• 
and  the  following  theorem  : 

Theorem.     If  in  the  leading  coefficient  of  any  covariant  9 
we  make  the  replacements 

a0,  av  av         «3, 

,o/(/  (l/'riih-  lit  ii  properly  chosen  power  of  d>}(  —  f)  we  h<w>  mi 
expression  for  <f>  as  a  rational  Junction  of  the  set  of  m  associated 
forms 

9l(=/),  B,(=  0),  i?2,  i?3,  .... 


FUNDAMENTAL   SYSTEMS  161 

For  illustration  let  m  =  3,  /  being  a  binary  cubic.     Let  <f> 
be  the  invariant  R.     Then  since 

B2  =  (ab)(ac)bxex  .  axbxcx  =  ±(abfaxbx4  =  £  A  ./,  Bz  =fQ, 

where  A  is  the  Hessian,  and  Q  the  cubic  covariant  of  /,  the 
typical  representation  of/  is 

If  one  selects  for  </>  the  invariant 

-  I  R  =  (a0a3  -  fl^)2-  4(a0a2  -  afX^Og  -  a§), 

and  substitutes 

/~2,     0,     1A/-2,     -(?f-2/ 

JS  =  [(/-4^)2  +  ^/-8A3]/6 

That  IS,  _  Rp  =CyQ2  +  A3_ 


tiiere  results  -i  ^>g 


This  is  the  syzygy  connecting  the  members  of  the  funda- 
mental system  of  the  cubic/  (cf.  Chap.  IV,  §  4).  Thus  the 
expression  of  R  in  terms  of  the  associated  forms  leads  to  a 
known  syzygy. 


CHAPTER   VII 

COMBINANTS   AND  RATIONAL   CURVES 
SECTION   1.     COMBINANTS 

In  recent  years  marked  advances  have  been  made  in  that 
1) ranch  of  algebraic  invariant  theory  known  as  the  theory  of 
combinants. 

I.  Definition.  Let/,  g,  A,  •••  be  a  set  of  m  binary  forms  of 
order  n,  and  suppose  that  m  <  n  ; 

/  =  atf%  H ,  g  =  bQx\  +  .-.,  h  =  cQx'{  +  •  •-. 

Let 


<K%  «!>•••;  *c 


;   Xyi  x%) 


be  a  simultaneous  concomitant  of  the  set.     If  0  is  such  a 
function  that  when/,  g,  h,  •  ••  are  replaced  by 


the  following  relation  holds  : 

<£(a('r  «j,  •••  ;   &{,,  ...  ;   cj,  •••  ;   a^,  a2) 

=  (£?£— )*<K«o'  «ii  •"  '  Jc  —  5   No- 
where 

£i<  VV     fc 

S2'  ^2'       '2' 
53"        VS'        531 


(160) 


•rr  r2),        (161) 


then  $  is  called  a  combinant  of  the  set  (Sylvester). 

We  have  seen  that  a  covariant  of  /  in  the  ordinary  sense 
is  an  invariant  function  under  two  linear  groups  of  trans- 

162 


COMBINANTS   AND   RATIONAL   CURVES         163 

formations.  These  are  the  group  given  by  T  and  the  in- 
duced group  (23j)  on  the  coefficients.  A  combinant  is  not 
only  invariantive  under  these  two  groups  but  also  under  a 
third  group  given  by  the  relations 

4=^1  +  ^1  +  £ici+  — * 

(162) 


As  an  illustration  of  a  class  of  combinants  we  may  note 
that  all  transvectants  of  odd  index  of/ and  g  are  combinants 
of  these  forms.     Indeed 

=  Qv)(f,gYr+\ 

by  (79)  and  (81).  Hence  (/,  #)2r+1  is  a  combinant.  In- 
cluded in  the  class  (163)  is  the  Jacobian  of  /and  g,  and  the 
bilinear  invariant  of  two  forms  of  odd  order  (Chap.  Ill,  V). 

II.  Theorem.  Every  concomitant,  cf),  of  the  set  f,  g,  h,  ••• 
ivhich  is  annihilated  by  each  one  of  the  complete  system  of  Aron- 
holoVs  polar  operators 

\db~r    \dc/     V   def    \   da/  '" 
is  a  combinant  of  the  set. 

Observe  first  that  <f>  is  homogeneous,  and  in  consequence 

where  ix  is  the  partial  degree  of  </>  in  the  coefficients  a  of  / 
i2  the  degree  of  <f>  in  the   coefficients    of  g,  and    so    forth. 


164  THE   THEORY   OF  INVARIANTS 

Since  fa -^U  =  0,  then  fa' -^W  =  0.     Thus 


3(£2«0  +  *?2J0  +  •"  +°Vo) 


+  (£i«i  +  i?A+-+«Vi)- 


3(f2al  +  i/2*l  +  ••'  +  o^l)      (164) 


+  (f !«.  +  9A  +  •  •  •  +  «,en)  f  -—-  =  0. 


M O....^    ^, *£. 


=  0. 


(165) 

1  £*«(&«*+  -+<Vt)               3£2 
+ 

+  axy *£ a(fa*<+---+<y,o=a0.      (166) 

gw+  ...  +cr2f't.)  5o-2 

Hence 

and  generally, 

^       \gs  d£t  dVt  *       datr   j  =  itf  (,  =  0,    (167) 

where  i  is  the  total  degree  of  <£>  in  all  of  the  coefficients.     In 

(167)  we  have  m2  equations  given  by  (s,  t=l,  ••■,  ?»).      We 

select  the  following  m  of  these  and  solve  them  for  the  deriv- 

d<f>' 
atives-2-,  ••• : 

«.  36'  3d>'  3(6'       n 

3?l  a7?i  30-j 


*  ^  +  7?  M  +  ...+,  ^  =  0 

3?i  3?/1  ao-j 


COMBLNANTS   AND   RATIONAL   CURVES         165 
Solution  of  these  linear  equations  gives 

djP  di)  dp 

Bat  we  know  that 

#-=^^1+M(?,/1+...+^'(fel. 

Hence 

Hence  we  can  separate  the  variables  and  integrate: 

d<f>'  _  .  dD 
$  ~h  2)' 

<f>'  =D>iF(a0,  ...),  (169) 

where  F  is  the  constant  of  integration.  To  determine  F, 
particularize  the  relations  (162)  by  taking  all  coefficients 
f,  77,  •••  zero  except 

£1  =  ^2=  •'•  =  °V»  —  1. 
Then  a{,  =  a0,  aj,  =  «r  •  ••,  b[  =  6f,  etc.,  and  (169)  becomes 

4>  =  F. 
Hence  <f>' =D^<f>, 

which  proves  the  theorem. 

It  is  to  be  noted  that  the  set  (168)  may  be  chosen  so  that 
the  differentiations  are  all  taken  with  respect  to  ffc,  %,  •••  in 
(168).     Then  we  obtain  in  like  manner 

Thus  ?\  =  L=  •••  =  im. 


166 


THE   THEORY   OF   INVARIANTS 


That  is,  a  combinant  is  such  a  simultaneous  concomitant  that 
its  partial  degrees  in  the  coefficients  of  the  several  forms  are 
all  equal.     This  may  be  proved  independently  as  the 

III.    Theorem.     A  combinant  is  of  equal  partial  degrees  in 
the  coefficients  of  each  form  of  the  set. 

We  have 


Hence 


aif){hla)-{hh 


a±)Jb± 

da       \    Ob 


<t>  =  Q. 


<j>={ix  -t2)<£  =  0. 

Thus  ix  =  i2.     Similarly  i7-  =  ik  (y,  k  =  1,  2,  •  •-,  m). 

IV.  Theorem.  The  resultant  of  two  binary  forms  of  the 
same  order  is  a  combinant. 

Let  f=f(xv  x2),  g  =  g(xv  x2). 

Suppose  the  roots  of  /  are  (rj",  r2iy)  (i  =  l,  •••,  w),  and  of  g 
(sj°,  s2l))  (i=  1,  •••  n).     Then  the  resultant  may  be  indicated 

by 

R  =  g(r?\  r$>)g(r<*\  rf )  •••  g(r["\  r<2" >), 
and  by 

B  =/(»(!>,  a^O/Oi2',  sf)  ...f(s["\  4»»). 

Hence 

(a£jR  =  2f(r^\  r^)g(rf.  rf>)  - g(r?\  r«»>)  =  0, 

(*>£\R  =  ZgCsi",  s!V)f(s?\  sf )  ■••/«>,  4«>)  =  0. 

Thus  7?  is  a  combinant  by  Theorem  II. 

Gordan  has  shown*  that  there  exists  a  fundamental  combi- 
nant of  a  set  of  forms.  A  fundamental  combinant  is  one  of 
a  set  which  has  the  property  that  its  fundamental  system  of 

*  Mathematische  Anrialen,  Vol.  5. 


COMBINANTS   AND   RATIONAL   CURVES 


167 


concomitants  forms  a  fundamental  system  of  combinants 
of  the  set  of  forms.  The  proof  of  the  Theorem  II  of  this 
section  really  proves  also  that  every  combinant  is  a  homo- 
geneous function  of  the  determinants  of  order  wi, 


K 


h 


that  can  be  formed  from  the  coefficients  of  the  forms  of  the 
set.  This  also  follows  from  (162).  For  the  combinant  is  a 
simultaneous  invariant  of  the  linear  forms 

f«*  +  vh  +  &k  4-  —  +  <rlk     <Jc=  0,  1,  ••  •,  w),         (170) 

and  every  such  invariant  is  a  function  of  the  determinants 
of  sets  of  m  such  linear  forms.  Indeed  if  we  make  the 
substitutions 

i  =  ii*' +&/  +  •■• +  &»*', 

V  =  Vi?  +  V2V'  H \-  Vm<r', 


in  (170)  we  obtain 


K  =  tiak  +  yJ>k  +  Lek  + 


and  these  are  precisely  the  equations  (162). 
For  illustration,  if  the  set  of  #-ics  consists  of 

g  =  b0xj  +  2  b^x^  +  ^l 

any  combinant  of  the  set  is  a  function  of  the  three  second 
order  determinants 

(a^j  —  ajJ0),  (a0b2  —  a260),  (a^  —  a2b{). 

Now  the  Jacobian  of  /  and  g  is 


168  THE   THEORY  OF   INVARIANTS 

Hence  any  corabinant  is  a  concomitant  of  this  Jacobian. 
In  other  words  J  is  the  fundamental  combinant  for  two 
quadratics.  The  fundamental  system  of  combinants  here 
consists  of  J  and  its  discriminant.  The  latter  is  also  the 
resultant  of/ and  g. 

The  fundamental  system  of  combinants  of  two  cubics  /",  g, 
is  (Gordan) 

#  =  (/,>),  0  =  (f,gy,  A  =  (/>,  *)»,  (*,  *)*    (A,  #),  (A,  *)«. 

The  fundamental  combinants  are  #  and  #,  the  fundamental 
system  consisting  of  the  invariant  0  and  the  system  of  the 
quartic  #  (cf.  Table  II). 

V.  Bezout's  form  of  the  resultant.     Let  the  forms  /,  g  be 
quartics, 

f=«<A  +  a\x\x2+  '"» 
9  =  Vi  +  h$¥%  +  —  • 

From/=  0,  #  =  0  we  obtain,  by  division, 

a0  _  a^f  4-  a^x\x2  +  ^ayl  +  ^4^1 
^0      Kxl  +  *2-rfr2  +  Vl2!  +  ^4 ' 
rt0a;1  +  q^-g  _  (?2:Kf  +  fl^i^  +  aAx% 
hxi  +  V2     hxi  +  hxix2  +  Ml' 

tf0.r|  +  q^'^2  +  ^2-Yl  _  a3yl  +  ^4^2 
^Vl  +  ^l-rl-r2  +  ^2-r2         ^3X1  +  ^4^2 

a0x^  +  axx\x2  +  a2xxx\  +  fl^aaj  _  «4 
?>0a-f  +  ^i^iJ'2  +  b<ix\x\  +  ^3-rl       ^4 


Now  we  clear  of  fractions  in  each  equation  and  write 


We  then  form  the  eliminant  of  the  resulting  four  homoge- 
neous cubic  forms.  This  is  the  resultant,  and  it  takes  the 
form 


COMBINANTS   AND   RATIONAL   CURVES 


169 


Poi 

P02 

Pm 

Pu 

Po2 

Pos  +  Pw 

Po4+Pl3 

Pu 

^03 

Poi  +  Pw 

Pu+Pzz 

P24 

^04 

Pu 

Pu 

Pu 

E  = 


Thus  the  resultant  is  exhibited  as  a  function  of  the  deter- 
minants of  the  type  peculiar  to  combinants.  This  result  is 
due  to  Bezout,  and  the  method  to  Cauchy. 

SECTION  2.     RATIONAL   CURVES 

If  the  coordinates  of  the  points  of  a  plane  curve  are 
rational  integral  functions  of  a  parameter  the  curve  is  called 
a  rational  curve.  We  may  adopt  a  homogeneous  parameter 
and  write  the  parametric  equations  of  a  plane  quartic  curve 
in  the  form 

^i  =  «io^i  +  anfi?2  +  •••  +  a14f|  =/i  (ir  f2), 

x2  =  a2of  1  +  «2l£l£2  +    •  •  •    +  «24^2  =  A  (f  1>  f  2>  (170i) 

*3  =  «30ll  +  a3lll?2  +   ••'    +  <*U%2  =/3  (&'   &)' 

We  refer  to  this  curve  as  the  i24,  and  to  the  rational  plane 
curve  of  order  n  as  the  Rn. 

I.  Meyer's  translation  principle.  Let  us  intersect  the 
curve  jR4  by  two  lines 

Ux  =  WjiEj  +  u2x2  +  MgZJg  =  0, 
vx  =  y^  +  v2.v2  -f  y3^3  =  0. 

The  binary  forms  whose  roots  give  the  two  tetrads  of  inter- 
sections are 

uf  =  (a10Mj  +  a20u2  +  «30«3  )%\  +  (a^  +  a21u2  +  a31M3)|?|2 

+  («i2Wi  +  «22W2  +  aWM8)SS+(a18Ul+aa8Ma  +  a88M8)fl^ 
+  (  aUUl  +  a24W2  +  aUUa)(& 

and  the  corresponding  quartic  vf.  A  root  (£[°,  fi,0)  of 
uf  =  0  substituted  in  (lTOj)  gives  one  of  the  intersections 
(af\  4°,  4°)  of  wx=  0  and  the  i24. 


170 


THE    THEORY    OF   INVARIANTS 


Now  Uf  =  0,  vf  =  0  will  have  a  common  root  if  their  result- 
ant vanishes.  Consider  this  resultant  in  the  Bezout  form 
R.     We  then  have,  by  taking 

a*  =  "h"i  4-  OmW2  4-  auuz     (i=  0,  •  ••,  4), 

Pik  =  aiuakv  ~   aivaku- 

Thus 

Pik  =(uv)x(  a^ask  -  a&aK )  +  (  uv  ).2(  a3iau  -  a^) 

where(i<v)1  =  w2v3— ».,?'.2.  (  ^v)2  =  M3t;1  — WjVg,  (uv}3=u1v2  —  u2vr 

1 1 1 -nee 


Pft  = 


(UV)1  (MV)2  ('"'); 


a2t 

«2ft 


«3i 


But  if  we  solve  w^.  =  0,  i>x  =  0  we  obtain 

x1  :  x2:  x3  =  (uv')1  :  ( iiv)2  :  (wv)3- 


Therefore 


Pik  =  ° 


•'i 
a, 


^2 


K        «3 


(t,A=rO,  ....  4), 


&lJfc       a2fc       #3fc 

where  o-  is  a  constant  proportionality  factor.     We  abbreviate 

pik  =  a\xaiak\. 

Now  substitute  these  forms  of  pu.  in  the  resultant  R.  The 
result  is  a  ternary  form  in  xv  x2,  xs  whose  coefficients  are 
functions  of  the  coefficients  of  the  Rr  Moreover  the  vanish- 
ing' of  the  resulting  ternary  form  is  evidently  the  condition 
that  ux  =  0,  vx  =  0  intersect  on  the  R±.  That  is,  this  ternary 
form  is  the  cartesian  equation  of  the  rational  curve.  Similar 
results  hold  true  for  the  Rn  as  an  easy  extension  shows. 

Again  every  combinant  of  two  forms  of  the  same  order  is 
a  function  of  the  determinants 


Pik  = 


a{     ak 
h     hk 


COMBINANTS   AND   RATIONAL   CURVES         171 

Hence  the  substitution 

pik  =  a  |  xa{ak  |, 

made  in  any  combinant  gives  a  plane  curve.  This  curve  is 
covariantive  under  ternary  collineations,  and  is  called  a  co- 
variant  curve.  It  is  the  locus  of  the  intersection  of  ux  =  0, 
vx  =  0  when  these  two  lines  move  so  as  to  intersect  the 
rational  curve  in  two  point  ranges  having  the  projective 
property  represented  by  the  vanishing  of  the  combinant  in 
which  the  substitutions  are  made. 

II.    Covariant  curves.   For  example  two  cubics 

/=  «0.rf  +  axx\x2  +  ••.,  g  =  bQx\  +  b^p^  +  •••, 
have  the  combinant 

K=  (a063  -  «A)-iOA  -  a2hi)- 

When  K=  0  the  cubics  are  said  to  be  apolar.  The  rational 
curve  Rz  has,  then,  the  covariant  curve 

K(x)  =  |  xa0a3  \  —  -J  |  xaxa2  \  =  0. 

This  is  a  straight  line.  It  is  the  locus  of  the  point  (wx,  vx~) 
when  the  lines  ux  =0,  vx  =  0  move  so  as  to  cut  Rz  in  apolar 
point  ranges.  It  is,  in  fact,  the  line  which  contains  the  three 
inflections  of  M3,  and  a  proof  of  this  theorem  is  given  below. 
Other  theorems  on  covariant  curves  may  be  found  in  W.  Fr. 
Meyer's  Apolaritat  und  Rationale  Curven  (1883).  The 
process  of  passing  from  a  binary  combinant  to  a  ternary 
covariant  here  illustrated  is  called  a  translation  principle. 
It  is  easy  to  demonstrate  directly  that  all  curves  obtained 
from  combinants  by  this  principle  are  covariant  curves. 

Theorem.  The  line  K(x)  =  0  passes  through  all  of  the 
inflexions  of  the  rational  cubic  curve  M3. 

To  prove  this  we  first  show  that  if  g  is  the  cube  of  one  of 
the  linear  factors  of /=  a(,P a(c2)  a(?\ 

g  =  (a[1>x1  +  <4%2)3, 


172  THE   THEORY   OF   INVARIANTS 

then  the  corabinant  K  vanishes  identically.     In  fact  we  then 
have 

and  a0  =  a[vafaf,  ax  =  So^o*2^,  .... 

When  these  are  substituted  in  TTit  vanishes  identically. 

Now  assume  that  ux  is  tangent  to  the  Rz  at  an  inflexion 
and  that  vx  passes  through  this  inflexion.  Then  uf  is  the 
cube  of  one  of  the  linear  factors  of  vf,  and  hence  K(x) 
vanishes,  as  above.  Hence  K(x)  =  0  passes  through  all 
inflexions. 

The  bilinear  invariant  of  two  binary  forms  /,  g  of  odd 
order  2  n  +  1  =  m  is 

Km  =  aQbm  -  ma1bn^.J  +  (o  W»»-2  +  •••  +  mam-A  -  amb0, 


or 


where /=  a0x1l  +  ?na1x7^~1x2  +  •••• 

If  two  lines  ux  =  0,  vx  =  0  cut  a  rational  curve  Rm  of 
order  m=2«+l  in  two  ranges  given  by  the  respective 
binary  forms 

uf,  Vf, 

of  order  m,  then  in  order  that  these  ranges  may  have  the 
projective  property  Km  =  0  it  is  necessary  and  sufficient  that 
the  point  (wz,  vx~)  trace  the  line 


^oo^tc-iy^r^o 


This  line  contains  all  points  on  the  Rm  where  the  tangent 
has  m  points  in  common  with  the  curve  at  the  point  of 
tangency.  The  proof  of  this  theorem  is  a  direct  extension 
of  that  above  for  the  case  m  =  3,  and  is  evidently  accomplished 
with  the  proof  of  the  following : 


COMBINANTS   AND   RATIONAL   CURVES         173 

Theorem.     A  binary  form,  f  of  order  m  is  apolar  to  each 
one  of  the  m,  m-th  powers  of  its  own  linear  factors. 
Let  the  quantic  be 

m 

f=  a™  =  a0z?  +  ...  =  IlOf  ^  -  r[J\). 
j=i 

The  condition  for  apolarity  of/ with  any  form  g  —  bf  is 

(ab)m  =  a0bm  -  mafi^  +...+(-  l)maj0  =  (f,  g)m  =  0. 
But  if  g  is  the  perfect  m-th  power, 

g  =  (r^xx  —  r^x2)m  =  (xrU))m, 
we  have  (cf.  (88)) 

which  vanishes  because  (r[j),  r^})  is  a  root  of  /. 

To  derive  another  .type  of  combinant,  let/,  g  be  two  binary 
quartics, 

/=  a0x\  +  ±axx\xt  H .     g  =  b0x\  +  -ib^fx^  H . 

Then  the  quartic  F=f+kg  =  A0x\  +  •••,  has  the  coefficient 

At  =  at  +  kb{     (i  =  0,  1,  -..,  4). 

The  second   degree  invariant   iF  =  A0A^  — 4AlA3  +  SAl  of 
F  now  takes  the  form 

B2i 
i  +  Si  -k  +  j-rk2  =  iF, 

where  S  is  the  Aronhold  operator 

and 

i  ==  a0a4  —  4  axaz  +  3  a\. 

The  discriminant  of  iF,  e.g., 

G  =  (Siy-2i(S?i), 


174  THE   THEORY  OF  INVARIANTS 

is  a  combinant  of  the  two  quartics/,  g.     Explicitly, 

^  =  Poi  +  1QPlS  ~   SPosPu  ~  SPoiPu  +  12A^24  -  48 ^12^23- 

Applying  the  translation  principle  to   6r  =  0  Ave    have    the 
covariant  curve 

(r(./-)=   'VV  2  +  "Hi  "l"sX\2  —  l\a0a3x\\alaAX\~  11^0*1^11*8^1 

+  ||  aQa2x  |  \a2a^x  |  —  j-2\aia2x\  \a2asx\ =  0- 

If  iF  =  0  the  qnartic  F  is  said  to  be  self-apolar,  and  the 
curve  Gr(x)  =  0  has  the  property  that  any  tangent  to  it  cuts 
the  BA  in  a  self-apolar  range  of  points. 


CHAPTER   VITI 

SEMINVARIANTS.     MODULAR   INVARIANTS 
SECTION  1.     BINARY   SEMINVARIANTS 

We  have  already  called  attention,  in  Chapter  I,  Section  1, 
VIII,  to  the  fact  that  a  complete  group  of  transformations 
may  be  built  up  by  combination  of  several  particular  types 
of  transformations. 

I.   Generators  of  the  group  of  binary  collineations.     The 

infinite  group  given  by  the  transformations  T  is  obtainable 
by  combination  of  the  following  particular  linear  transfor- 
mations : 

t:xl  =  \x,  x2  =  ny, 

t^.x^x'  +  vy>,  y  =  y'., 
t2  :  x'  =  x'v  y'  =  ax[  +  x'2. 

For  this  succession  of  three  transformations  combines  into 

2^  =  51(1-+  (jv)x\  +  \vx'2, 
x2  —  <t\xx\  +  /t.r2, 

and  evidently  the  four  parameters, 

\x2  =  /la,   X2  =  aft,   ixx  =  Xf,   \1  =  X(l  +  cry), 

are  independent.     Hence  the  combination  of  t,  tv  t2  is 

T:x1  =  \lx[  +  fi^,  x2  =  \x[  +  ix2x'2. 

In  Section  4  of  Chapter  VI  some  attention  was  given  to 
fundamental  systems  of  invariants  and  covariants  when  a 
form  is  subjected  to  special  groups  of  transformations  Tp. 
These  are  the  formal  modular  concomitants.  Booleans  are 
also  of  this  character.     We  now  develop  the  theory  of  in- 

175 


176  THE   THEORY   OF  INVARIANTS 

variants  of  a  binary  form  /  subject  to  the  special  transfor- 
mations tr 

II.  Definition.  Any  homogeneous,  isobaric  function  of 
the  coefficients  of  a  binary  form  f  whose  coefficients  are 
arbitrary  variables,  which  is  left  invariant  when  /  is  sub- 
jected to  the  transformation  tx  is  called  a  seminvariant.  Any 
such  function  left  invariant  by  t2  is  called  an  anti-semin- 
variant. 

In  Section  2  of  Chapter  I  it  was  proved  that  a  necessary 
and  sufficient  condition  that  a  homogeneous  function  of  the 
coefficients  of  a  form  f  of  order  m  be  an  invariant  is  that  it 
be  annihilated  by 

0  =  mal h  (m  —  1)«2^ h  •••  +  '*,, 


dax  da2  dam 

We  now  prove  the  following  : 

III.  Theorem.  A  necessary  and  sufficient  condition  in  order 
that  a  function  i",  homogeneous  and  isobaric  in  the  coefficients 
of  /=«'",  may  be  a  seminvariant  of  f  is  that  it  satisfy  the 
linear  partial  differential  equation  £11=  0. 

Transformation  of  /=  a^x™ -\-  malx1^~lx2  +  •••  by  tx  gives 
/'  =a'0x'{'  +  ma\x'{n-'i.ii  -\ ,  where 

a\  =  rtj  +  a0v, 

a'2  =  a2+2  ajy  +  aQv2, 


<  =  am  +  mam_xv  +  (  ™  W-2*1  +  •••  +  a0V 


Hence 


— -"  —  u,   — ->._a0,   — -« av   — >  —  o  a2,  •••,  — — —  w«m_j. 

dy  dt>  oi'  dy  oz/ 


SEMINVARIANTS.     MODULAR   INVARIANTS      177 

Now  we  have 

dl(al  a'r-..)=  dIBa!n  |    BI  Ba\   (  3J  da'm 

Bv  da'0  Bv        Ba\   Bv  Ba'm   Bv 

But  — ^  '*'         =  0  is  a  necessary  and  sufficient  condition  in 
Bv 

order  that  /(«[>,  •••,  «4)  may  be  free  from  v,  i.e.  in  order  that 

I(a'0,  •••)  may  be  unaffected  when  we  make  v  =  0.     But  when 

v  =  0,  a'j  =  cij  and 

/(>(,,  -..,  a'm)  =  I(a0,  ...,  am~). 

Hence  —  =  £1'  (a'0,  ...)=  0  is  the  condition  that  I(a!^  •••)  be 

a  seminvariant.  Dropping  primes,  ill  (a0,  ...)  =  0  is  a  nec- 
essary and  sufficient  condition  that  /(a0,  •••)  be  a  sem- 
invariant. 

IV.  Formation  of  seminvariants.  We  may  employ  the 
operator  O  advantageously  in  order  to  construct  the  sem- 
invariants of  given  degree  and  weight.  For  illustration  let 
the  degree  be  2  and  the  weight  w.  If  w  is  even  every  sem- 
invariant must  be  of  the  form 

1=  a0alc  +  \lalaw_l+  X2a2a((,_2  +  •••  +  X^a^. 

Then  by  the  preceding  theorem 

£11=  (w  +  \1)a0aw_1+((w—  l)Xj  +  2\2)a1rtM,_2H —  =  0. 

Or 

w  +  \x  =  0,  (w  -  l)Xj  +  2  X2  =  0,  O  -  2)X2  +  3  X3  =  0,  ..., 
(iw+l)X.w_1  +  wX.w  =  0. 

Solution  of  these  linear  equations  for  \v  X2,  •••  gives 
1=  a0aw  -H Ja^.j  +( %]a2al0_2 


178  THE   THEOEY   OF   INVARIANTS 

Thus  there  is  a  single  seminvariant  of  degree  2  for  every 
even  weight  not  exceeding  m. 

For  an  odd  weight  w  we  would  assume 

1=  ((o"<r  +  \alaw-l  +  •"  +  ^i(w-l)a|f»-l)%w+l)' 

Then  £11=  0  gives 

w  +  \  =  0,  (w  -  l)\j  +  2  X2  =  0,  •-., 
i(w  +  3)\i(JC_3,  +  Uiv  -  1  )Xi(w_1}  =  0,  \h,c_v  =  0. 

Hence  Xx  =  X2  =  •••  =  Xi((r_1,=  0,  and  no  seminvariant  exists. 
Thus  the    complete  set    of   seminvariants  of   the  second 
degree  is 

A0  =  a% 

A2  =  a0a2  —  of, 

A±  =  a0a4  —  4  axa^  +  3  a|, 

,A6  =  a0aG  —  6  a^g  +  15  a2aA  —  10  a§, 

^48  =  a0a8  —  8  a^  +  28  a2a6  —  56  a3«5  +  35  a\. 

The  same  method  may  be  employed  for  the  seminvariants 
of  any  degree  and  weight.  If  the  number  of  linear  equa- 
tions obtained  from  £11=  0  for  the  determination  is  just 
sufficient  for  the  determination  of  \v  X2,  X3,  •••  and  if  these 
equations  are  consistent,  then  there  is  just  one  seminvariant 
/of  the  given  degree  and  weight.  If  the  equations  are  in- 
consistent, save  for  X0=  Xx  =  X0  =  •••  =0,  there  is  no  semin- 
variant. If  the  number  of  linear  equations  is  such  that  one 
can  merely  express  all  X's  in  terms  of  r  independent  ones,  then 
the  result  of  eliminating  all  possible  X's  from  /  is  an 
expression 

1  =  X^  +  X272  +  ...  +  Xr7r. 

In  this  case  there  are  r  linearly  independent  seminvariants 
of  the  given  degree  and  weight.     These  may  be  chosen  as 


SEMINVARIANTS.     MODULAR   INVARIANTS      179 

V.  Roberts'  theorem.  If  0Q  is  the  leading  coefficient  of  a 
covariant  of  f=  aQx'^  -f-  ■••of  order  w,  and  CM  is  its  last  coeffi- 
cient, then  the  covariant  may  be  expressed  in  the  forms 

OP  0'2(y  0">  O 

6W  +  -TTX1      x2  "I n**!     x\  +   •••    H Q*o,  (173) 

— 7— S«i'  +  -1 fa?  %  -f  •..  +—-j±Xix%  x+  Ojjq.       (174) 

|ft>  |G>  —   1  |1 

Moreover,  0Q  is  a  seminvariant  and  0M  an  anti-semi  nvariant. 
Let  the  explicit  form  of  the  covariant  be 

K=  CQx?+(j)01a%-%+  ...  +  C>£. 

Then  by  Chapter  I,  Section  2,  XII, 

Cl-x2—)K==0. 
dxx 

Or 


+  o)(n  C,,.!  -  w  -  1  CUz)^-1  +  (fl  (7W  -  tw  C^)^  =  0. 

Hence  the  separate  coefficients  in  the  latter  equation  must 
vanish,  and  therefore 

O  Cft  =  0, 


The  first  of  these  shows  that  C0  is  a  seminvariant.  Combin- 
ing the  remaining  ones,  beginning  with  the  last,  we  have  at 
once  the  determination  of  the  coefficients  indicated  in  (174). 


180  THE   THEORY   OF  INVARIANTS 

In  a  similar  manner 


{°-*>£)K=0> 


and  this  leads  to 

OC0  =  a>Cv  OC1  =  (a)  -  1)  G>,  •  ••,  067_i  =  Om,  OC„=0; 

&)(o)-l)(G)-2)...  (»-t  +  l)        oV 

This  gives  (173). 

It  is  evident  from  this  remarkable  theorem  that  a  co- 
variant  of  a  form  /  is  completely  and  uniquely  determined 
by  its  leading  coefficient.  Thus  in  view  of  a  converse 
theorem  in  the  next  paragraph  the  problem  of  determining 
covariants  is  really  reduced  to  the  one  of  determining  its 
seminvariants,  and  from  certain  points  of  view  the  latter  is 
a  much  simpler  problem.  To  give  an  elementary  illustration 
let/ be  a  cubic.     Then 

ft       s  ^     i   o         d  3 

0  =  6  a, h  1  a2 f-  a3 , 

and  if  O0  is  the  seminvariant  a0a2  —  a\  we  have 

OC0  =  a0a8  -  «jrt2,  O2C0  =  2(rtla3  -  a|),  O3^  =  0. 
Then  2  if  is  the  Hessian  of  /,  and  is  determined  uniquely 


VI.  Symbolical  representation  of  seminvariants.  The  sym- 
bolical representation  of  the  seminvariant  leading  coefficient 
C0  of  any  co variant  if  of  /,  i.e. 

K=(ab)p(ac)"  ■■■  </>V    •••  (r  +  s  +  t  +  ■■■  =  a>), 

is  easily  found.      For,  this  is  the  coefficient  of  xx  in  if,  and  in 
the  expansion  of 

(aby(ac)"  ..•  (a^  +  a^X&i2!  +  V2)*  "• 


SEMINVARIANTS.     MODULAR  INVARIANTS      181 

the  coefficient  of  x"  is  evidently  the  same  as  the  whole  ex- 
pression K  except  that  ax  replaces  ax,  bx  replaces  bx,  and  so 
forth.     Hence  the  seminvariant  leader  of  .ST  is 

C0  =  (aby(acY  ...  a$\c\  ....  (175) 

(r  +  s  +  t  +  •••  a  positive  number). 

In  any  particular  case  this  may  be  easily  computed  in  terms 
of  the  actual  coefficients  of  /  (cf.  Chap.  Ill,  §  2,  I). 

Theorem.  Every  rational  integral  seminvariant  of  f  may 
be  represented  as  a  polynomial  in  expressions  of  the  type  (70, 
with  constant  coefficients. 

For  let  <f>  be  the  seminvariant  and 

<K«!p  — )=<K«0'  "0 
the  seminvariant  relation.     The  transformed  of 
/  =  {a1xl  +  a2x2)m 

f  =  [axx[  +  (axv  +  «2)4]m. 

If  the  #0,  «j,  •••  in  <£(a0,  •••)  are  replaced  by  their  symbolical 
equivalents  it  becomes  a  polynomial  in  av  «2,  /3r  /32,  •••  say 
F(av  «2,  /3r  /32,  •••).     Then 

=  ^T(a1,  «2,  /Sj,  /32,  •••). 
Expansion  by  Taylor's  theorem  gives 

<^+^4+7i4+ ■•■)2?(ai'  *  *•■ ft-  -)=o- 

Now  a  necessary  and  sufficient  condition  that  F  should  sat- 
isfy the  linear  partial  differential  relation 


by 
is 


182  THE   THEORY   OF   INVARIANTS 

is  that  F  should  involve  the  letters  a^  /32,  •••  only  in  the 
combinations 

(a/3),  (ay),  (/97),  •••• 

In  fact,  treating  BF  =  0  as  a  linear  equation  with  constant 
coefficients  (ax,  /3r  •••  being  unaltered  under  ^)  we  have  the 
auxiliary  equations 

da2  _  d/32  _  dy2  _         _  dF 
«i        &i         7i  ° 

Hence  F  is  a  function  of  («/3),  (07).  •••  with  constant  coeffi- 
cients which  may  involve  the  constants  av  /3V  •••.  In  other 
words,  since  (f>(a0)=F(av  •••)  is  rational  and  integral  in  the 
a's  F  is  a  polynomial  in  these  combinations  with  coefficients 
which  are  algebraical  rational  expressions  in  the  av  ftv  •••. 
Also  every  term  of  such  an  expression  is  invariant  under  tv 
i.e.  under 

a\  =  Oj,  «2  =  axv  +  «2,   •••, 

and  is  of  the  form 

r0  =  («/3)H«7)p'"-«?/3i  ..-, 
required  by  the  theorem. 

We  may  also  prove  as  follows  :  Assume  that  J7  is  a  func- 
tion of  (a/3),  (a7),  (aS),  •••  and  of  any  other  arbitrary 
quantity  s.     Then 

dF  dF      d(a/3)   ,  dF      8(  ay)   ,  dF  ds 

a,  ——  =  a, —      v  +  a, — - — '-£■  +  •  •  •  +  a, , 

da%  d(a(3)       o«2  d(ay)       d«2  ds  da2 

dF  Q  dF  dQc/3)  B  dF  d(gy)  ,Rd_F^s_ 
Pl  <9/32  Pl  d(a/3)  d£2  Pl  d(ay)  d/32  *"  Pl  a*  a/32' 
etc. ,  •     • 

T>    f  dF      d(a/3)  o       dF 

But  a, v  ^    =  —  «,/?, , 

^(a/S)       ^  *    ^(a/S) 

,  _9^_  ac«£)_       s  _dJ^_ 

*d(«/3)     3/8a    -  +  "l/"13(«^)' 


SEMINVARIANTS.     MODULAR   INVARIANTS      183 
Hence  by  summing  the  above  equations  we  have 

Since  s  is  entirely  arbitrary  we  can  select  it  so  that  8s  =£  0. 

n  IT 

Then  —  =0,  and  F,  being  free  from  s,  is  a  function  of  the 

ds 

required  combinations  only. 

Theorem.  Every  seminvariant  off  of  the  rational  integral 
type  is  the  leading  coefficient  of  a  covariant  off. 

It  is  only  required  to  prove  that  for  the  terms  T0  above 
w  =  p  +  a  +  •••  is  constant,  and  each  index 

p,  <r,   ••• 

is  always  a  positive  integer  or  zero.     For  if  this  be  true  the 
substitution   of    ar,  /3j.,  •••  for    av  /3V  •••   respectively  in  the 
factors  a^fil  -..of  ro  and  the  other  terms  of  F,  gives  a  co- 
variant  of  order  a>  whose  leading  coefficient  is  $(«0,  •••)• 
We  have 

If  the  degree  of  </>  is  i,  the  number  of  symbols  involved  in  ro 
is  i  and  its  degree  in  these  symbols  im.  The  number  of 
determinant  factors  (a/3)  •••  is,  in  general, 

w=pl+p2+  •••  +pm-v, 

and  this  is  the  weight  of  <f>.  The  degree  in  the  symbols  con- 
tributed to  T0  by  the  factors  (a/3)  ...is  evidently  2  w,  and  we 
have  p,  er,  •••all  positive  and 

im  >  2  w, 
that  is, 

o)  =  im  —  2w  >0. 
For  a  more  comprehensive  proof  let 

i="»4+/3'4+"" 


184  THE   THEORY   OF   INVARIANTS 

Then 

Hence,  since  T0  is  homogeneous  in  the  symbols  we  have  by 
Euler's  theorem, 

(Bd  -  dB)T0  =  O  +  co  -  w)r0  =  a>r0, 
(Bd2  -  dB2)T0  =  (Bd  -  dB)dT0  +  d(8d-  dB)T0  =  2(g>-  l)dro, 

(  8d*  -  rfS*)P0  =  &(>  -  *  +  l)^"1^  (jfe  =  1,  2,  •••), 

But 


Also 


8T0  =  0,  hence  BdkT0  =  k(co-k  +  l)^*-1^. 

da{  =  da^ai^  =  (m  —  i)aM  =  Oa{  (i  =0,1,  •••,  m  —  1), 

d<f>  =  -2-  da0  +  -^  <Za,  +  ■••  +  — -^-^am_1  =  0</>. 
3a0  dax  dam_x 

Hence  dkT0  is  of  weight  w  +  k. 
Then 

(/'m-w+T0  =  0. 

For  this  is  of  weight  im  +  1  whereas  the  greatest  possible 
weight  of  an  expression  of  degree  i  is  im,  the  weight  of  a'm. 
Now  assume  <o  to  be  negative.     Then  dim~lcT0  =  0  because 

Bdim-W+1T0  =  (im  -  w  +  1)  [a>  -  (im  -  w  +  1)+  l]dim-tT0  =  0. 

Next  ^im-"-1r0  =  0  because 

Bdim-'rT0  =  (m  -  w)  [a>  -  (iw  -  w)  +  l]rf'm-!r-T0  =  0. 

Proceeding  in  this  way  we  obtain  T0  =  0,  contrary  to  hy- 
pothesis.    Hence  the  theorem  is  proved. 

VII.    Systems  of  binary  semin variants.     If  the  binary  form 
f  =  a()x™  4-  ma-ffi~xx%  +  •••  be  transformed  by 

#1  =  .Fj  +  v%2?  2*2  =  "*V 


SEMINVARIANTS.     MODULAR   INVARIANTS      185 

there  will  result, 

/'  =  Off  +  m  C^-%  +  h)c2x^-2x'^  +  • . .  +  02™, 

in  which 

C{  =  a0v{  +  ia^-1  +  (  o)<Vi-2  +  •••  +  ^-iy  +  «i-     (176) 

Since  £IC0  =  fl«0  =  0,  0Q  is  a  semin variant.  Under  what  cir- 
cumstances will  all  of  the  coefficients  Qi  (i  =  0,  •••,  m)  be 
semin  variants  ?     If  Cx  is  a  seminvariant 

£10-^  =  D,(a0v  +  aj)  =  aoni>  +  a0  =  0. 

That  is,  £lv  =  —  1.  We  proceed  to  show  that  if  this  condi- 
tion is  satisfied  fl  Ot  =  0  for  all  values  of  i. 

Assume  D,v  =  —  1  and  operate  upon  Ct  by  H.     The  result 
is  capable  of  simplification  by- 

Hit5  =  8i/-1fiy  =  —  8v*~\ 
and  is 

0(7<=-»vM-(i)(<-1)«i*,M (**)(*-r)fl^-ta-1— - 

-  Ww  +  ( j)  «o^'_1  +  2  (2)  V~2  +  '  " 

+(r  +  \)(r  +  IH*^1  +  -  +  *Vi- 

But 

f    *    V    ,  n     *(*  —  1)  •••  (*  —  r +  !)(%  —  r)     fi\,.       . 

(vr+i/r+i>=—     — p — - — -=yo-o. 

Hence  II (7,-  =  0. 

Now  one  value  of  v  for  which  £lv  =  —  1  is  v  = l  •     If  /  be 

transformed  by 

"^l  —     1  2'      2  —     2' 

then  Cj  =  0,  and  all  of  the  remaining  coefficients  C{  are  sem- 
in variants.     Moreover,  in  the  result  of  the  transformation, 


186  THE   THEORY   OF   INVARIANTS 

r{=  a1'1 0{  =  aj_1«j  —  (  J  )ao~2«i-ilh  +11  )«o~3a*-2ai  ~~  ' " 
+  (  -  l)^jWfl^  +  (  -  ly-1^  -  1  )a* 


This  gives  the  explicit  form  of  the  seminvariants.     The  trans- 
formed form  itself  may  now  be  written 

f>  —  F    r'"'  4-  f  m\       2  r''«-2r'2    i    [™1  ^  3  r'»-3r'3  _|_   ...   _l_      ^  '"     r'm 

J  —L<Fi  +\-2)y  l     -     wfa  J     2         +  p*-i  2  • 

Theorem.  Every  seminvariant  of  f  is  expressible  ration- 
ally in  terms  of  ro,  T2,  T3,  •••  ,  r,„.      0/<f  obtains  this  expression 

r 

%  replacing  ax  by  0,  a0  by  T0,  and  ciff^  0,  1)   oy  — J-  *Vi  the 

0 
original  form  of  the    seminvariant.      Except  for  a  power  of 

ro  =  «0    in  the  denominator  the  seminvariant  is  rational  and 
integral  in  the  T'i(i  =  0,  2,  •••,  m)  (Cayley). 

In  order  to  prove  this  theorem  we  need  only  note  that/' 
is  the  transformed  form  of  /  under  a  transformation  of  de- 
terminant unity  and  that  the  seminvariant,  as  S,  is  invarian- 
tive  under  this  transformation.     Hence 

S(T*  °'  £"'  Pi'  -'  T^l)=  ^  ^'  a*  -'  a"^     (177) 
^  1010  i0/ 

which  proves  the  theorem. 

For  illustration  consider  the  seminvariant 

S  =  a0a4  —  4  fljflg  -f-  3  a|. 
This  becomes 

s=-^(3ri  +  r4), 

ro 

CI- 
AS' =  a0a4  —  4  rtjrtg  +  3  a\ 

=  —,  [3(  <roa2 _  a?)2+  (4a4 —  4  «daia3  +  6  v^ _  %  *!)]■ 

«0 


SEMINVARIANTS.     MODULAR   INVARIANTS      187 


This  is  an  identity.      If   the    coefficients   appertain    to   the 
binary  quartic  the  equation  becomes  (cf.  (125)) 

Again  if  we  take  for  *S'  the  cubic  invariant  J  of  the  quartic 
we  obtain 

0 


H= 


ir 


0 


ir 

a?    3 


0 


or 


1  asJ- 

6  "(r 


x  21  4         x  2         x  3' 

Combining  the  two  results  for  i  and  J"  we  have 

rr  =i  a2iT  —  3  T3  —  i  «3,/_l  rs  _i_  T2 

1  21  4         2  "0tx  2        °  x  2  —  6  "O"7  +  x  2  ^  x  3* 

Now  2  T2  is  the  sem  in  variant  leading  coefficient  of  the 
Hessian  J3"of  the  quartic/,  and  F3  is  the  leader  of  the  co- 
variant  T.  In  view  of  Roberts'  theorem  we  may  expect  the 
several  covariants  of  /  to  satisfy  the  same  identity  as  their 
seminvariant  leaders.  Substituting  |  H  for  T2,  T  for  T3, 
and/  for  a0,  the  last  equation  gives 

if3  +  lf3J+  2T2~}2  if*H=  0, 
which  is  the  known  syzygy  (cf.  (140)). 

SECTION  2.     TERNARY   SEMINVARIANTS 

We  treat  next  the  seminvariants  of  ternary  forms.      Let 
the  ternary  quantic  of  order  m  be 

^  I m 


l\    |  "'2  |  "'3 

When  this  is  transformed  by  ternary  collineations, 
xx  =  XjZj  +  /"i-*^  +  vix3"> 
V:     x2  =  \2x\  +/a2x2  +  v2xv 

x3  =  \x[  +  fx3x[  +  v3x!,,    (Xfiv}^  0, 


188  THE   THEORY   OF   INVARIANTS 

it  becomes/',  where  the  new  coefficients  a'  are  of  order  m  in 
the  Vs,  /x's,  and  vs.  This  form  /  may  be  represented  sym- 
bolically by 

/  =  a™  =  (a^  +  a2r2  +  a3.r3)m. 
The  transformed  form  is  then  (cf.  (76)) 

/'  =  (aKx\  +  a^%  +  at,4)m  (178) 

Then  we  have 


//  n"h/i'"-'/i"la 


Now  let 


d\         d  d  d 


Then,  evidently  (cf.  (75)  and  (23^) 
\m  (     d  Y'V    d  Y"3 


W2  —  W2  —  w& 

(179) 
This  shows  that  the  leading  coefficient  of  the  transformed 
form  is  a™,  i.e.  the  form  /  itself  with  (x)  replaced  by  (X), 
and  that  the  general  coefficient  results  from  the  double 
ternary  polarization  of  a™  as  indicated  by  (179). 

Definition.     Let  <£  be  a  rational,  integral,  homogeneous 
function  of  the  coefficients  of/,  and  <}>'  the  same  function  of 

the    coefficients  of  /'.     Then    if   for   any   operator   (a4^—  \ 
( \—  ],  •••,  say  for  ( X — J,  the  relation 

is  true,  <f>  is  called  a  semin variant  of/. 


SEMINVARIANTS.     MODULAR   INVARIANTS      189 

The  reader  should  compare  this  definition  with  the  ana- 
lytical definition  of  an  invariant,  of  Chapter  I,  Section  2,  XL 

I.  Annihilators.  A  consequence  of  this  definition  is  that 
a  seminvariant  satisfies  a  linear  partial  differential  equation, 
or  annihilator,  analogous  to  £1  in  the  binary  theory. 

For, 


dfij  da'„m\      dp  J  da'        \ 


da' 


d/A 


MLJ\  da'w™ 


and 


da'oom\      dP  f 


1  '  *  —  1  K q-  )a\ xaM 2<*v 3  —  miaK '    «V 2    «„ 3  —  w2ami+1  ^.j OTg. 
Hence 

V    ^  $  5«W3  (180) 

Now  since  the  operator 


2JW2< 


+lmjs-l«nsn    / 

7/ij  "'»l'»2m3 

annihilates  </>'  then  the  following  operator,  which  is  ordinarily 
indicated  by  fl^^,  is  an  annihilator  of  <f>. 

°*»  =  T»VWl  hh-I  ms  n Ol  +  ™2  +  ™3  =  ™).  (181) 

The  explicit  form  of  a  ternary  cubic  is 

f~  ^300^1  "^  ^  a2102*i-r2  "t"  "  a\2fP\X2  "^  a030r2  ~^  "  aW>lXlXZ 

In  this  particular  case 

#j  -^  «j  *j 

^x,i.  =  «300  J" 1"  ^  «2io  77         f"  3  «120  T"         H  a201 


*<%„        21"a«120  •««■»      ""Sam 

°"021  u'<012 


190  THE   THEORY   OF   IX  V  Alii  ANTS 

This    operator    is    the    one  which  is  analogous  to  £1  in  the 
binary  theory.     From  f  n—  !(/>',  by  like  processes,  one  obtains 

the  analogue  of  0,  e.g.  £lx^.  Similarly  12^,  O^,  O^,  fi^ 
may  all  be  derived.  An  independent  set  of  these  six  opera- 
tors characterize  full  invariants  in  the  ternary  theory,  in  the 
same  sense  that  H,  0  characterize  binary  invariants.  For 
such  we  may  choose  the  cyclic  set  £lXlxtt  A/V>v  ^r.tjy 
Now  let  the  ternary  m-\c  form 

+  m(am_101xf-1  +(»»  -  l)am_2113-'1"-2.r2  -\ \-  ,/„„,   n./-.r  -'  )./3 

+ , 

be  transformed  by  the  following  substitutions  of  determinant 
unity : 

a~  =  asL  (183) 


Then  the  transformed  form/'  lacks  the  terms  x'/'^Kii,  .r["'~\r'.,. 
The  coefficients  of  the  remaining  terms  are  semin variants. 
We  shall  illustrate  this  merely.     Let  m  =  2, 

f=  a20Qp1  +  2  «no-ri-r2  +  rto2o-r2  +  *  aioia;ia;3  "^  ""  aon2:2a;3  +  a(mX3' 
Then 

^200.'      =  ^200^1      '     (a020a200  —  aho)X2"  "*"  -v'oila200  —  'rHil''llo ^-r2*r3 

It  is  easy  to  show  that  all  coefficients  of/'  are  annihilated  by 

a    . 

Likewise  if  the  ternary  cubic  be  transformed  by 

x\  —  xl  x2  ■'  3' 

a300  "300 


SEMINVARIANTS.     MODULAR   INVARIANTS      191 

and  the  result  indicated  by  ag00/'  =  A3Qtfc'f  +  3  ^vp\x%  +  '"' 
we  have 

^300  =  4)0'  (184> 

^-210  =  ^' 

-^•120  =  rt300Vrt300al20  —  a210/' 

-^030  =  "  a210  ~"  °  a210^120a300  "^  a030a300' 

^201  =  ^' 

■^•111  =  a30oCa300aill  —  a210rt201  )' 

^021  —  rt30((a021  —  a300a201al20  —  Z  rt210rtlllrt300  "+"  -1  a210a201' 

■^•102  =  a300va300a102  —  a20l)'1 

^012  =  a300a012  —  a300a102a210  —  "  a300a20lalll  +  ^  ^Ol^lO1 

^003  =  ^  a201  —       a300a20lal02  "^"  a003a300* 

These  are  all  seminvariants  of  the  cubic.  It  will  be  noted 
that  the  vanishing  of  a  complete  set  of  seminvariants  of  this 
type  gives  a  (redundant)  set  of  sufficient  conditions  that  the 
form  be  a  perfect  with  power.  All  seminvariants  of  /  are 
expressible  rationally  in  terms  of  the  .A's,  since  /'  is  the 
transformed  of/by  a  transformation  of  determinant  unity. 

II.    Symmetric  functions  of  groups  of  letters.     If  we  mul- 
tiply together  the  three  linear  factors  of 

"  (afx1  +  afx2  +  afz3), 

the  result  is  a  ternary  cubic  form  (a  3-line),  /=  #300#i  +  •••. 
The  coefficients  of  this  quantic  are 

am  =  Sa^afof  =  a^afaf, 

«2io  =  2«pct[2>ag>  =  a^afaf  +  a™a™af  +  a^afaf, 

«120  =  ^^a^a™  =  a[va^af  +  a™afaf  +  a^afaf , 

am  =  2a2i>«22)a|3)  =  «<i>«<2>rt<3>, 

%>i  =  ^«i1,«i2'43)  =  «l1,«l2,«f  +  c^afaf  +  a!/]afaf, 


192  THE   THEORY   OF   INVARIANTS 

am  =  I,aWaU>af  =  a™al®af  +  «<!>««»  ««$)  +  «(i)a<.2>a(3>t 
a102  =  lu^'u^af  =  a'^ufaf  +  41)42)48)  +  aa  ^(2)  43^ 

a0i2  =  Sa^af  43)  =  a^^afaf  +  ^'af  «f  +  41,42>43', 
a003  =  2a(i)«(2)aC3)==aa)«(2)ac3). 

These  functions  2  are  all  unaltered  by  those  interchanges  of 
letters  which  have  the  effect  of  permuting  the  linear  factors 
of/ among  themselves.  Any  function  of  the  a\j)  having  this 
property  is  called  a  symmetric  function  of  the  three  groups 
of  three  homogeneous  letters, 

04",  41),  «a>), 

(«<2\    «f,    42>), 
(af),    43),    aC3)). 

In  general,  a  symmetric  function  of  m  groups  of  three  homo- 
geneous letters,  «r  «2.  «3.  i.e.  of  the  groups 

7l(4i\  41),  41)), 

72(42\     «22\     42)), 


is  such  a  function  as  is  left  unaltered  by  all  of  the  permuta- 
tions of  the  letters  a  which  have  the  effect  of  permuting  the 
groups  7r  72,  •••,  ym  among  themselves:  at  least  by  such  per- 
mutations. This  is  evidently  such  a  function  as  is  left  un- 
changed by  all  permutations  of  the  superscripts  of  the  «'s. 
A  symmetric  function  of  m  groups  of  the  three  letters 
uv  «2,  «3,  every  term  of  which  involves  as  a  factor  one  each 
of  the  symbols  a(1),  «(2\  •••,  «("°  is  called  an  elementary  sym- 
metric function.  Thus  the  set  of  functions  a3W  a2l0,  •••  above 
is  the  complete  set  of  elementary  symmetric  functions  of 
three  groups  of  three  homogeneous  variables.  The  non- 
homogeneous  elementary  symmetric  functions  are  obtained 
from  these  by  replacing  the  symbols  41}>  a32)>  a33)  eacn  by 
unity. 


SEMINVARIANTS.     MODULAR   INVARIANTS      193 

The  number  iV  of  elementary  symmetric  functions  of  m 
groups  of  two  non-homogeneous  variables  «ro, 0, 0,  <V_i,i,o?  ••• 
is,  by  the  analogy  with  the  coefficients  of  a  linearly  factorable 
ternary  form  of  order  m, 

N.=  m  +  m  +  (m  -  1)  +  (m  -  2)  +  ...  +2  +  1  =  1  m(m  +  3). 

The  N  equations  aiik  =  2,  regarded  as  equations  in  the  2  m 
unknowns  a[r),  a(2s)  (r,  8=1,  •••,  m),  can,  theoretically,  be  com- 
bined so  as  to  eliminate  these  2  m  unknowns.  The  result  of 
this  elimination  will  be  a  set  of 

\  m(m  +  3)  —  2  m  =  \  m(m  —  1) 

equations  of  condition  connecting  the  quantities  am0oi 
am_110,  •••  only.  If  these  a's  are  considered  to  be  coefficients 
of  the  general  ternary  form  /  of  order  m,  whose  leading  co- 
efficient a003  is  unity,  the  \m(m  —  1)  equations  of  condition 
constitute  a  set  of  necessary  and  sufficient  conditions  in  order 
that/ may  be  linearly  factorable. 

Analogously  to  the  circumstances  in  the  binary  case,  it  is 
true  as  a  theorem  that  any  symmetric  function  of  m  groups 
of  two  non-homogeneous  variables  is  rationally  and  integrally 
expressible  in  terms  of  the  elementary  symmetric  functions. 
Tables  giving  these  expressions  for  all  functions  of  weights 
1  to  6  inclusive  were  published  by  Junker  *  in  1897. 

III.  Semi-discriminants.  We  shall  now  derive  a  class  of 
seminvariants  whose  vanishing  gives  a  set  of  conditions  in 
order  that  the  ternary  form /of  order  m  may  be  the  product 
of  m  linear  forms. 

The  present  method  leads  to  a  set  of  conditional  relations 
containing  the  exact  minimum  number  \m(m—  1)  ;  that  is, 
it  leads  to  a  set  of  \m(m  —  1)  independent  seminvariants  of 
the  form,  whose  simultaneous  vanishing  gives  necessary  and 
sufficient  conditions  for  the  factorability.  We  shall  call 
these  seminvariants  semi-discriminants  of  the  form.     They 

*  Wiener  Denksehrif ten  for  1897. 


194  THE   THEORY  OF  IXVARIA^TS 

are  all  of  the  same  degree  2  m  —  1  ;  and  are  readily  formed 
for  any  order  m  as  .simultaneous  invariants  of  a  certain  set  of 
binary  qualities  related  to  the  original  ternary  form. 

If  a  polynomial,  f3m)  of  order  ra,  and  homogeneous  in  three 
variables  (xv  xv  x3~)  is  factorable  into  linear  factors,  its  terms 
in  (xv  £2)  must  furnish  the  (xv  x2~)  terms  of  those  factors. 
Call  these  terms  collectively  a[£,  and  the  terms  linear  in  x3 
collectively  x3a™r~~l.  Then  if  the  factors  of  the  former  were 
known,  and  were  distinct,  say 

<= «ooii<  Tfx\  -  *■!%)+  n  ofo. 

the  second  would  give  by  rational  means  the  terms  in  x3  re- 
quired to  complete  the  several  factors.  For  we  could  find 
rationally  the  numerators  of  the  partial  fractions  in  the 
decomposition  of  a'{'~^ /«'",.,  viz. 

m 

m_l         11  r2      m 

a\r      —    i=\  "V*  ai 

and  the  factors  of  the  complete  form  will  be,  of  course, 


rPx-,  —  ri^x, 


{l)x2+  a{x3     (»asl,  2,  ..-,  m). 


Further,  the  coefficients  of  all  other  terms  in  fSm  are  rational 
integral  functions  of  the  r(i)  on  the  one  hand,  and  of  the  at 
on  the  other,  symmetrical  in  the  sets  (r£\  —  r['\  a,).  We 
shall  show  in  general  that  all  these  coefficients  in  the  case 
of  any  linearly  factorable  form  are  rationally  expressible  in 
terras  of  those  occurring  in  a'^,  a'f/K  Hence  will  follow  the 
important  theorem, 

Theorem.  If  a  ternary  form  fSm  is  decomposable  into  linear 
factors,  all  its  coefficients,  after  certain  -  m,  are  expressible 
rot  tonally  in  terms  of  those  2  m  coefficients.  That  is,  in  the 
space  ivhose  coordinates  are  all  the  coefficients  of  ternary  forms 
of  order  m,  the  forms  composed  of  linear  factors  fill  a  rational 
spread  of  2  m  dimensions. 


SEMINVARIANTS.     MODULAR   INVARIANTS      195 

We  shall  thus  obtain  the  explicit  form  of  the  general 
ternary  quantic  which  is  factorable  into  linear  factors. 
Moreover,  in  case  f3m  is  not  factorable  a  similar  development 
will  give  the  theorem, 

Theorem.  Every  ternary  form  f3m,  for  which  the  discrimi- 
nant D  of  afr  does  not  vanish,  can  be  expressed  as  the  sum  of 
the  product  of  m  distinct  linear  forms,  plus  the  square  of  an 
arbitrarily  chosen  linear  form,  multiplied  by  a  "  satellite  "  form 
of  order  m  —  2  whose  coefficients  are,  except  for  the  factor  D~l, 
integral  rational  seminvariants  of  the  original  form  f3m. 

A  Class  of  Ternary  Seminvariants 

Let  us  write  the  general  ternary  quantic  in  homogeneous 
variables  as  follows  : 

J 'Am  =  a0x  "T"  &\x     #3  +  #2.r     #3  "f"    •"    *+"  ^mfr^S  ' 

where 

oJT*  =  (hP^~*  +  a^r*"1^  +  a^-^xl  +  •..  +  aim^x^ 

(i  =  0,  1,  2,  •••,  m). 
Then  write 


,m-l  sim-l 


«r; «iir ^        «* 


2ml  rL&)x~  — 


"''  TTrr'«r    _  r<*>r  ^        *=1    2       *  *       2 

11  V2    Xl         rl    X2>*  (n       _r(pr(2)   ...  r(»0\  . 

and   we    have    in    consequence,    assuming   that   D  =£  0,  and 
writing 


hn 
«0r<*>   ~ 


'da'n 


u  te^ 


..1=,.  (<>,,,=,< 


w 


5<rl  -W«      . 

-T '    a0r(lc)  — 

_        -c  1   J-'l— 'l      I   -'2—'  2 

the  results 

«*  =  r^fl^,/«6«  =  -  K*Xr»/C*>-  (185) 

Hence  also 

a''™ '   =-?!i_<i'«}  (186) 

^2 


196  THE   THEORY   OF  INVARIANTS 

The  discriminant  of  agj.  can  be  expressed  in  the  following 
form : 

m 

^>  =  II*>/a00(-1)ini(m-1).  d87) 

,y=i 
and  therefore 


r2    ah.a-)a0ra)%r(2)  '"  (V(*-i)(Va-+i)  •••  a0r(m) 


(188) 


a00(-  l)J'»(«-"i) 
and  in  like  manner  we  get 

m 

n«x-  =  <%&%£  ••'  aiS/(-  l)im(m-l)i>.  (189) 

The  numerator  of  the  right-hand  member  of  this  last  equal- 
ity is  evidently  the  resultant  (say  Rm)  of  afx  and  af~l. 
Consider  next  the  two  differential  operators 

1  *)  *) 

Al  =  ^«00T~  +  (m  ~  1)a01  5 +  •••   +  <hm-\ 


•     \---  -  /  " 01   i  '  '         vm  —  i.  i  i 

<3a10  dan  da^^ 

a  3        ,   /  in  d        ,  .  d 

A2  =  7wa0jB- hO-l)a0m-i7 1 l-fl01- — ; 

3alOT_!  3alm_2  da10 

and  particularly  their  effect  when  applied  to  a"].-1.     We  get 
(cf.  (186)) 


Cfl 


/  2 

and  from  these  relations  we  deduce  the  following  : 

ATI  a  A*Rm  -  a    vSW^5l^      nan 

or,  from  (185) 

^&l =2a1a2  •••  am_iHm).  (192) 

(_  ]\im(TO-l)  J  2)  "  " 

In  (191)  the  symmetric  function  2  is  to  be  read  with  refer- 
ence to  the  r's,  the  superscripts  of  the  r's  replacing  the  sub- 
scripts usual  in  a  symmetric  function.  Let  us  now  operate 
with  A2  on  both  members  of  (191).     This  gives 

. — .  7—z — —  —   Mnn^ 


f        1  \*m(jn-l)  1 1    /)  u0       n'm     n'm         ..n'm  V         r(m-\)         ,.(»i) 


SEMINVARIANTS.     MODULAR   INVARIANTS      197 

Let  SA  represent  an  elementary  symmetric  function  of  the 
two  groups  of  homogeneous  variables  rv  r2  which  involves 
h  distinct  letters  of  each  group,  viz.  r[m~i+l\j  =  1,  2,  •  ••,  h). 
Then  we  have 

^*Rm        -  =  2[(-  1)^  ...  am^2r['»-Vr<>» >].   (193) 


(_i)i»(»-i)|i|2i) 

We  are  now  in  position  to  prove  by  induction  the  follow 
ing  fundamental  formula  : 


A£~'~'&R« 


(  _  l)*m<m-D  |w_  8_  ^j)  (194) 

=  2[(-  l)^  •••  «52m_/>f+1Vf+2)  ...  r[s+l)r(2s+M)  •••  r<"°] 
(s  =  0,  1,  •••,  m;  t  ass  0, 1,  •••,  m  —  s), 

where  the  outer  summation  covers  all  subscripts  from  1  to 
wV superscripts  of  the  r's  counting  as  subscripts  in  the  sym- 
metric function.  Representing  by  Jm_s_^t  the  left-hand 
member  of  this  equality  we  have  from  (190) 

A    7"  -Sir       n(+l\'"\'2'         ai,.(*-D 

iltf/  m_s_t,  t—*\K     XJ      ,m    ,m        -fa 

r(s)  \ 

V  ^(1)^(2)  .    .  ~(s)  11_V         ~<s+l)  ..     ~(s+t) r(s+t+l)  ...  r(m)  ), 

A   '  2    '2             '2        («)»»—*' 1  '1           2                     '2       I 

^*2  ' 

This  equals 

2(-iy+V2  •••«*-!# 

where  #  is  a  symmetric  function  each  term  of  which  involves 
t  -\- 1  letters  ^  and  m  —  s  —  t  letters  r2.  The  number  of 
terms  in  an  elementary  symmetric  function  of  any  number 
of  groups  of  homogeneous  variables  equals  the  number  of 
permutations  of  the  letters  occurring  in  any  one  term  when 
the  subscripts  (here  superscripts)  are  removed.  Hence  the 
number  of  terms  in  SM_4  is 

\m  —  s 


in 


& 


and  the  number  of  terms  in  S  is 

(m  —  s  -f  1 )  |  m  —  s/ !  1 1  m  —  s  —  t . 


198  THE   THEORY   OF   INVARIANTS 

But  the  number  of  terms  in 

^m—s+lK'  l    'l  '1        r2  '2     ) 

is 

m  —  s  +  l/[  m  —  s  —  t\t  +  1. 
Hence 

£  =  (£  +  l)2m_,+1, 
and  so 

£    ■     1 LV         1  -1        '*1<JC2  ••'  «*-i  —  OT-jr— lJ* 

This  result,  with  (193),  completes  the  inductive  proof  of 
formula  (194). 

Now  the  functions  Jm-s-t,t  are  evidently  simultaneous  in- 
variants of  the  binary  forms  <r,  ajf,  a£\  ajpi.  We  shall 
show  in  the  next  paragraph  that  the  expressions 

/„,_,_,, <  =  Dast  -  DJm_s_u  t    (*  =  2,  3,  » .,  w ;  £  =  0,  1, ....  w  -  «) 

are,  in  reality,  semin variants  of  the  form/.),,,  as  a  whole. 

Structure  of  a  Ternary  Form 

The  structure  of  the  right-hand  member  of  the  equality 
(194)  shows  at  once  that  the  general  (factorable  or  non-fac- 
torable) quantic  fSm{D  =£  0)  can  be  reduced  to  the  following 
form  : 

111  '"      W> — «? 

/*■ = n  (r*)xi  - r^ + «*> + s  2 ( a*< -  J'"-s-1- *>  **~*~%  • 

*=i  .=,-  *=o  (195) 

This  gives  explicitly  the  "satellite"  form  of  f:im.  with  coeffi- 
cients expressed  rationally  in  terms  of  the  coefficients  of/3m. 
It  may  be  written 

-  iym(m-l)\m  —  8  —  t\tj 


Df*m-2  =  X  Xl1***'  ~  r_-\\h 


J  2 


m      III  —  V 

=  XXI»>-s-t,lz,rs~'4-  (i96) 


SEMLNVARIANTS.     MODULAR   INVARIANTS      199 

Now  the  coefficients  Im_s_ut  are  semin variants  of  f3m.     To 
fix  ideas  let  m=S  and  write  the  usual  set  of  ternary  operators, 

<H)0  5«01  3«02  5«10  5«11  5«20 

0^=3%,—-  +  2a01— -  +  a02~—  +  2a10- — '+a  u- — +  « 20-— , 
5^01  5a02  da03  5au  a«12       ^9a21 

daso  da20  da10       Llda21         "ldan       mda12 

etc. 

Then  I1Q  is  annihilated  by  £1.^  but  not  by  flr^,  ^x  is  anni- 
hilated by  D,Xl&  but  not  by  il.Vl,  and  J00  is  annihilated  by 
Ha:^  but  not  by  n^3.ri.  In  general  Im_s_ut  fails  of  annihilation 
when  operated  upon  by  a  general  operator  Hr.r.  which  con- 
tains a  partial  derivative  with  respect  to  ast.  We  have  now 
proved  the  second  theorem. 


The  Semi-discriminants 

A  necessary  and  sufficient  condition  that  f3m  should  de- 
generate into  the  product  of  m  distinct  linear  factors  is  that 
/t*m-2  should  vanish  identically.  Hence,  since  the  number  of 
coefficients  in  fxm_2  is  \  m(m  —  1),  these  equated  to  zero  give 
a  minimum  set  of  conditions  in  order  that/3m  should  be  fac- 
torable in  the  manner  stated.  As  previously  indicated  we 
refer  to  these  seminvariants  as  a  set  of  semi-discriminants  of 
the  form/3m.     They  are 

^'  "      (-l^™-»\t\m-8-t[t=0,  1,  -.,  m-sj  (      } 

They  are  obviously  independent  since  each  one  contains  a 
coefficient  (a8t)  not  contained  in  any  other.  They  are  free 
from  adventitious  factors,  and  each  one  is  of  degree  2  m  —  1. 


200 


THE    THEORY   OF   INVARIANTS 


In  the  case  where  m  =  2  we  have 


*±ac 


aoo 

«01 

a( 

00 

la 

-«02 

+ 

«10 

an 

0 

111 

0 

<X-.U 

a. 

-Ax)  —        a20 

This  is  also  the  ordinary  discriminant  of  the  ternary 
quadratic. 

The  three  semi-discriminants  of  the  ternary  cubic  are  given 
in  Table  V.  In  this  table  we  have  adopted  the  following- 
simpler  notation  for  the  coefficients  off: 

/  =  aQx\  +  «r4r2  +  azxv4  +  a$4. 

+  'CI- 
TABLE   V 


Tid 

-7m 

— /uo 

4  rtja2«3fro 

«1«3^0 

«360 

-«l^o 

—  3  a^b^ 

+  «l«360^f 

-9a|6g 

+  a\a8bl 

+  a\b<f>l 

+  3  (iiasbl 

—  3  a./>3b'i 

-  2  ff2506^ 

—  (ilbf 

+  a*b* 

-  aoaztfp^ 

+  3  a2b\ 

—  4  Ojfl^&l 

+  a2blbs 

-<%b\ 

+  9  azb\ 

—  2  a^^fto 

+  6  a1a3b0b2 

+  9  «3  V>1 

+  3  03&O^A 

—  2  a|ft0fc2 

—  «1a2a36061 

—  a1a2b0b1b2 

+  a^dzbj)^ 

+  2  a\a3b0b2 

-  O3&1 

+  3  a2«3^i&i 

—  6  a2asb0b2 

—  ai&id| 

-  4  ofagftoftj 

+  4  a^fc;. 

+  «26j62 

+  «1«2&1&2 

—  3  ajftgftjbo 

+  &I 

—  '.<  (^fe^o 

—  ttja^jft.. 

+  «i"'^,, 

—  aia2e0 

+  djafcj 

+  18  a,a2ffl3c?0 

—  18  rtia2a3c0 

+  18  a^a.MzCi 

—  4(/L//n 

+  4  fl'2r0 

—  4  af^ 

-4afa  ^ 

+  4  a?a3c0 

-  4  ofo,^ 

-  27  afd0 

+  27  «jjc0 

—  27  <73c, 

SEMINVARIANTS.     MODULAR   INVARIANTS      201 
In  the  notation  of  (197)  the  semin variants  in  this  table  are 

^10=-Z>a20  +  AA' 

J01  =  Da21  +  A2i23, 
where  D  is  the  discriminant  of 

and  M3  the  resultant  of  a  and 

/3  =  a10a?  +  anxxx2  +  «12a|. 
Corresponding  results  for  the  case  m  =  4  are  the  following  : 

where 

H  =  a02  —  "  rt0la03  +-'--'  a00a04' 

"  1  =  "  '  ftola04  ~t~  ^  '   *00ao3  "I"  "  ^02  —   '  ^  ^00*02^04  ~  ^  ^01^02*03 ' 
«10  all  rt12  a13         °        ° 

0  aw  an  an     a1B    0 

0  0  a10  an     a12  a18 

fyl^lO- a00all    a02a10 —  a00^12    a03<rl0— a00al3    ^04^10      ^        ^ 

a00  a01  a02  a03     a04     U 

0  aQ0  aQ1  a02     a03  a04 

the  other  members  of  the  set  being  obtained  by  operating 
upon  RA  with  powers  of  Av  A2 : 

da10  d(hi  da12  oa13 

a         i  d         0  9        0          9  9 

A2  =  4  a04- —  +  3  a03— -  +  2  a02- —  +  %  5 — » 

according  to  the  formula 

T  -  ,    r>      A{~^A2i?4          (s  =  2,   3,  4;    f=0,   1,   ..., 

1*-^*-a*U      U~s-t\t  4-«Y 


i?4  = 


202 


THE    THEORY    OF    INVARIANTS 


1 V.  Invariants  of  /n-lines.  The  factors  of  a";v  being  assumed 
distinct  we  can  always  solve  Im_s_L(=  0  for  asl,  the  result  be- 
ing obviously  rational  in  the  coefficients  occurring  in  «,',".,  a"*"1. 
This  proves  the  first  theorem  of  III  as  far  as  the  case  I>  =£  U  is 
concerned.  Moreover  by  carrying  the  resulting  values  of 
a8t(s  =  2,  3,  •••,  m  ;  ^  =  0,  1,  •  ••,  m  —  s)  back  into  f3m  we  get 
the  general  form  of  a  ternary  quantic  which  is  factorable  into 
linear  forms.  In  the  result  <',.  a"*1  are  perfectly  general 
(the  former,  however,  subject  to  the  negative  condition 
J)=£0),   whereas 

\m—j  f?  A'"— )'-i\     /? 

(  _  \\hm(m-l) J)am-j  =  L±L__J±m xm-j  _j_  ^l  ^2Jt"' j.>; 

W  —  /  —  1     1 


+ 


\m-J 


=  ^-h±  + 


(,/ =  2.  3,  -..,  m}'. 


Thus  the  ternary  form  representing  a  group  of  m  straight 
lines  in  the  plane,  or  in  other  words  the  form  representing 
an  m-line  is,  explicitly, 

7/i         m—j  \m—i—j\i  J? 

_l  j)-\(  _  i\i»«(»«-D yVo V  ^ 2    m3f-^-jzi,.        (198) 

j=2        i=0  ' ^  '- 

This  form,  regarded  as  a  linearly  factorable  form,  possesses 
an  invariant  theory,  closely  analogous  to  the  theory  of  binary 
invariants  in  terms  of  the  roots. 

If  we  write  a$x  =  .?:;/„, t  ,,.  a'},.  =  .rVXn  ,,,  (%,  =  1).  and  assume 
that  the  roots  of  l^  =  0  are  —  rv  —  rT  —  r3,  then  the  factored 
form  of  the  three-line  will  be,  by  the  partial  fraction  method 
of  III  (185), 

3 
/  =  ]^[  (  «1  +  >V2  -  h-rJl'o-r)' 
i  =  l 

Hence  the  invariant  representing  the  condition  that  the  3-line 
/  should  be  a  pencil  of  lines  is 

1      rx      h-rjli 
Q=  1     r„     h- 


Jl'o- 


h    r,    /,', 


o-r, 


SEMINVARIANTS.     MODULAR   INVARIANTS      203 

This  will  be  symmetric  in  the  quantities  rv  r2,  r3,  after  it  is 
divided  by  Vi2,  where  R  =  (rx  —  r2)2(  r2  —  r3)2(r3  —  t^)2  is 
the  discriminant  of  the  binary  cubic  a%x.  Expressing  the 
symmetric  function  Qx  =  Q/y/M  in  terms  of  the  coefficients 
of  «$_,.,  we  have 


(J1  —  I  «01«13        aoi*02all  "^"  l   a00a03ttll  —  "  a0la03al0  "I"  *  a02rt10 

This  is  the  simplest  full  invariant  of  an  m-linef. 


D  rt00(/02rt12* 


SECTION  3.     MODULAR   INVARIANTS   AND   COVARIANTS 

Heretofore,  in  connection  with  illustrations  of  invariants 
and  covariants  under  the  finite  modular  linear  group  repre- 
sented by  Tp,  we  have  assumed  that  the  coefficients  of  the 
forms  were  arbitrary  variables.  We  may,  however,  in  con- 
nection with  the  formal  modular  concomitants  of  the  linear 
form  given  in  Chapter  VI,  or  of  any  form  /  taken  simulta- 
neously with  L  and  Q,  regard  the  coefficients  of/  to  be  them- 
selves parameters  which  represent  positive  residues  of  the 
prime  number  p.  Let  /  be  such  a  modular  form,  and 
quadratic, 

/=  a^x\  +  2  atxtx^  +  a2x2- 

Let  jo  =3.  In  a  fundamental  system  of  formal  invariants 
and  covariants  modulo  3  of  /  we  may  now  reduce  all  expo- 
nents of  the  coefficients  below  3  by  Fermat's  theorem, 

af  =  a{  (mod  3)   (i=0,  1,  2). 

The  number  of  individuals  in  a  fundamental  system  of  /  is, 
on  account  of  these  reductions,  less  than  the  number  in  the 
case  where  the  a's  are  arbitrary  variables.  We  call  the  in- 
variants and  covariants  of  /,  where  the  a's  are  integral, 
modular  concomitants  (Dickson).  The  theory  of  modular 
invariants  and  covariants  has  been  extensively  developed. 


204 


THE   THEORY   OF   INVARIANTS 


In  particular  the  finiteness  of  the  totality  of  this  type  of  con- 
comitants for  any  form  or  system  of  forms  has  been  proved. 
The  proof  that  the  concomitants  of  a  quantic,  of  the  formal 
modular  type,  constitute  a  finite,  complete  system  has,  on  the 
contrary,  not  been  accomplished  up  to  the  present  (December, 
1914).  The  most  advantageous  method  for  evolving  funda- 
mental systems  of  modular  invariants  is  one  discovered  by 
Dickson  depending  essentially  upon  the  separation  of  the 
totality  of  forms/'  with  particular  integral  coefficients  modulo 
p  into  classes  such  that  all  forms  in  a  class  are  permuted 
among  themselves  by  the  transformations  of  the  modular 
group  given  by  Tp*  The  presentation  of  the  elements  of 
this  modern  theory  is  beyond  the  scope  of  this  book.  We 
shall,  however,  derive  by  the  transvection  process  the  funda- 
mental system  of  modular  concomitants  of  the  quadratic 
form  /,  modulo  3.  We  have  by  transvection  the  following 
results  (cf.  Appendix,  48,  p.  241): 

TABLE    VI 


Notation 

Trans- 

YECTANT 

1  lONCOMlTANT    (MOB    3) 

A 

C/,/)2 

a\  —  rtnCfo 

<1 

C/8,  <?)G 

"n"-  +  "ii"l'  +  <V'l  +  <l~la2  ~  °0  ~  °2 

L 

3                      3 
■''i-''l'  —  35l*2 

Q 

{{L,L)*L) 

**'i  ~r  XiXo  ~\~  XiXo  ~\~  *Co 

f 

altx\  +  2  a-yXjX   +  a&\ 

A 

if,  Q)* 

a0x{  +  a^x2  +  owl  +  a--'A- 

d 

(/8,  <?)5 

("u"l- 

"i  '-'i  +  ( ",.  -  «2) (of  +  a^XjXi  +  (rtf  - 

2\     ' 

Co 

c/2,  Qy 

(o2  +  a\ 

-  a<fl2)x\  +  ary{a0  +  a^x^:.  +  (of  ■•  a\  - 

-  a0a2)  ri 

Also  in  q  and  C1  we  may  make  the  reductions  a\  =  a{  (mod  3) 
(i=0,  1,  2).  We  now  give  a  proof  due  to  Dickson,  thai 
these  eight  forms  constitute  a  fundamental  system  of 
modular  invariants  and  covariants  of  /'. 


'•  Transactions  American  Math.  Society,  Vol.  10  (1909),  p   123. 


SEMINVARIANTS.     MODULAR   INVARIANTS      205 

Much  use  will  be  made,  in  this  proof,  of  the  reducible  in- 
variant 

1=  (og  -  1)0/2  _  l)(a|  _  i)  =  q2  +  A2  _  1  (mod  3^ 

In  fact  the  linearly  independent  invariants  of /are 

1,  A,  I,  q,  A2.  (f) 

Proceeding  to    the  proposed   proof,  we   require  the  sernin- 
variants  of/.     These  are  the  invariants  under 

xx  =  x[  +  x'2,  x2  =  x'2  (mod  3). 

These  transformations  replace /by/',  where 

a'0  =  a0,  a\  =  a0  +  ax,  a2  =  a0  —  «j  +  a2  (mod  3).  (£) 

Hence,  as  may  be  verified  easily,  the  following  functions  are 
all  seminvariants : 

a0,  «$,  a0A,  a0A2,  «2A,  5  =  ( a2  -  l)ar  (s) 

Theorem.  Any  modular  seminvariant  is  a  linear  homo- 
geneous function  of  the  eleven  linearly  independent  seminvari- 
ants (z),  (s). 

For,  after  subtracting  constant  multiples  of  these  eleven, 
it  remains  only  to  consider  a  seminvariant 

S  =  a-^a^a^  +  o^jfl^  +  asai  +  «4«f«l  +  cc^afa^  +  «6a2  +  /8a2,  +  7a2, 

in  which  «r  «2  are  linear  expressions  in  a2,  a0,  1  ;  and 
«3,  •••,  «6  are  linear  expressions  in  a0,  1  ;  while  the  coefficients 
of  these  linear  functions  and  /3,  7  are  constants  independent 
of  a0,  av  a2.  In  the  increment  to  S  under  the  above  induced 
transformations  (f)  on  the  a's  the  coefficient  of  axa\  is  —  «0a4, 
whence  «4  =  0.  Then  that  of  a\a2  is  0^=0;  then  that  of 
ara2  is  /3  —  «0«5,  whence  /3  =  «5  =  0  ;  then  that  of  a\  is 
—  «2  =  0  ;  then  that  of  a1  is  —  7  —  «0«6.  whence  7  =  «6  =  0. 
Now  8  =  a3av  whose  increment  is  «3«0,  whence  «3  =  0 
Hence  the  theorem  is  proved. 

Any  polynomial  in  A,  I,  q,  a{V  B  is  congruent  to  a  linear 


206  THE   THEORY   OF  INVARIANTS 

function  of  the  eleven  seminvariants  (i),  (s)   by  means  of 
the  relations 


(mod  3), 


(A)    lA  =  Iq  =  la 0=lB  =  qA  =  qB  =  a0B  =  0, 

AB  =  B,  «2A2  =  A2  +  a%A  -  A. 

aQq  =  a*A2-  a%  B*=A(1-  a$)\ 

together  with  ajj  =  a0,  A3  =  A  (mod  3). 

Now  we  may  readily  show  that  any  covariant,  K.  of  order 
6t  is  of  the  form  P  +  X(7,  where  C  is  a  covariant  of  order 
6  t  —  4  and  P  is  a  polynomial  in  the  eight  concomitants  in 
the  above  table  omitting /4.  For  the  leading  coefficient  of  a 
modular  covariant  is  a  modular  seminvariant.  And  if  t  is 
odd  the  covariants 

//•'.  /£',  G™,  0"    0'  an  invariant) 

have  as  coefficients  of  #* 

a0z,  i,  .8,  A  +  «2, 

respectively.  The  linear  combinations  of  the  latter  give  all 
of  the  seminvariants  (*),  (s).  Hence  if  we  subtract  from  K 
the  properly  chosen  linear  combination  the  term  in  x\  cancels 
and  the  result  has  the  factor  xr  But  the  only  covariants 
having  x2  as  a  factor  are  multiples  of  L.  Next  let  t  be  even. 
Then 

/*,  A/3',  iQf*-*,  QC*-\  ixQK  (f  =  \  *'  f-     ) 

Vtj  =  Z  A,  A2,  q.J 

have  as  coefficients  of  .r*\' 

a2,  a2A,  a0i,  B,  iv 

Lemma.  If  the  order  co  of  a  covariant  C  of  a  binary 
quadratic  form  modulo  3  is  not  divisible  by  3,  its  leading 
coefficient  &1  is  a  linear  homogeneous  function  of  the  semin- 
variants (i),  (s),  other  than  1,  /,  q. 

In  proof  of  this  lemma  we  have  under  the  transformation 


SEMIN VARIANTS.     MODULAR   INVARIANTS      207 
For  a  covariant  C  the  final  sum  equals 

where  a'0,  •••  are  given  by  the  above  induced  transformation 
on  the  as.     Hence 

S'1-S1  =  a>S(mod  3). 
Now  write  St  =  ka\a\a\  +  t  (t  of  degree  <  6), 

and  apply  the  induced  transformations.      We  have 

S[  =  ka^aQ  +  a1)2(a0  —  ax  4-  «2)2  +  £' 

=  kaf^a^r  +  a\  +  axa2  +  a\a^)  +  £', 

where  r  is  of  degree  3  and  t'  of  degree  <  6.     Hence 

a»S=  &(a0r  +  a2af  +  a%a1a^)  +  t'  —  t  (mod  3). 

Since  &>  is  prime  to  3,  S  is  of  degree  <  6.  Hence  #  does  not 
contain  the  term  a%a\a\,  which  occurs  in  2"  but  not  in  any 
other  seminvariant  (T),  (s).  Next  if  S  =  1  +  a,  where  a  is  a 
function  of  a0,  av  a2  without  a  constant  term,  IC  is  a  covari- 
ant C  with  <S"  =  J.  Finally  let  S=q+  a1  +  a^  +  «3A2  +  ^ 
where  t  is  a  constant  and  the  «f  are  functions  of  a0.     Then 

by  (A) 

qS  =  I-A2  +  1  +«!?, 

which  has  the  term  a\a\a\  (from  I).  The  lemma  is  now 
completely  proved. 

Now  consider  covariants    C  of    order  o>  =  6  t  +  2.      For  £ 
odd,  the  covariants 

have  as  coefficients  of  x\ 

a2,  a0,  A2  —  a2A  +  a§,  «0A  +  ao'  ^ 

respectively.  Linear  combinations  of  products  of  these  by 
invariants  give  the  sem invariants  (s)  and  A,  A2.  Hence,  by 
the  lemma,    C=P  +  LG\  where  P  is  a  polynomial  in  the 


208 


THE   THEORY   OF   INVARIANTS 


covariants  of  the  table  omitting /4.  For  t  even  the  co- 
variants 

fQ^PQ'-K  C2Q<,  Otf 

have  a0,  a2,,  A  +  afr  B  as  coefficients  of  x". 

Taking   up    next    covariants   0  of   order   &>  =  6 1  +  4,  the 
coefficients  of  a-"  in 

are,  respectively,  a0,  a$,  B,  A  —  a2A.  Linear  combinations 
of  their  products  by  invariants  give  all  sera  in  variants  not 
containing  1,  /,  q.  Hence  the  eight  concomitants  of  the 
table  form  a  fundamental  system  of  modular  concomitants 
of/  (modulo  3).  They  are  connected  by  the  following 
syzygies  : 

fCx  ee  2(A2  +  A)i,     /<72  ee  (1  +  A)/4  )         d 
G\  -  0\  =  (A  +  l)2/2    C71  — jgT4  =  A  £  J  v 

No  one  of  these  eight  concomitants  is  a  rational  integral 
function  of  the  remaining  seven.  To  prove  this  we  find 
their  expressions  for  five  special  sets  of  values  of  «0,  av  a2 
(in  fact,  those  giving  the  non-equivalent  (fs  under  the  group 
of  transformations  of  determinant  unity  modulo  3): 


/ 

A 

'J 

C\ 

'  c 

ft 

(1) 

0 

0 

0 

0 

0 

0 

(2) 

A. 

0 

-1 

0 

4 

A 

(3) 

-T2 
—  J  j 

0 

1 

0 

■'"\ 

A 

~  -'l 

(4) 

,2    i       2 

t£-\    ~T"  Xq 

- 1 

0 

0 

0 

A  +  A 

(6) 

-  ■'' !■'*:: 

1 

0 

0     ,          9 

-X1  +  .''[, 

x\  +  A 

xv4 

To  show  that  L  and  Q  are  not  functions  of  the  remaining 
concomitants  we  use  case  (1).  For  /4,  use  case  (4).  No 
linear  relation  holds  between/,  Or  C2  in  which  01  is  present, 
since  Cx  is  of  index  1,  while  /  C2  are  absolute  covariants. 
Now  f^kC2  by  case  (4);  C2=£kf  by  case  (5).  Next 
q^F(A)  by  (2)  and  (3)  ;  A^F(q~)  by  (4)  and  (5). 


CHAPTER   IX 

INVARIANTS    OF    TERNARY    FORMS 

In  this  chapter  we  shall  discuss  the  invariant  theory  of  the 
general  ternary  form 

/=  a™  =  b';>  =  ••-. 

Contrary  to  what  is  a  leading  characteristic  of  binary  forms, 
the  ternary/  is  not  linearly  factorable,  unless  indeed  it  is 
the  quantic  (198)  of  the  preceding  chapter.  Thus  /  repre- 
sents a  plane  curve  and  not  a  collection  of  linear  forms. 
This  fact  adds  both  richness  and  complexity  to  the  invariant 
theory  of  f.  The  symbolical  theory  is  in  some  ways  less 
adequate  for  the  ternary  case.  Nevertheless  this  method 
has  enabled  investigators  to  develop  an  extensive  theory 
of  plane  curves  with  remarkable  freedom  from  formal 
difficulties.* 

SECTION   1.     SYMBOLICAL   THEORY 

As  in  Section  2  of  Chapter  VIII,  let 

f(x)  =a%  =  0Vl  +  a2x%  +  a3.r3)'"  =  6™ 

Then  the  transformed  of  /  under  the  collineations  V  (Chap. 
VIII)  is 

/'  =  (aAx[  +  a/.2  +  avx'zy\  (199) 

I.  Polars  and  transvectants.     If    (>jv  i/2,  ?/3)  is  a  set  co- 
gredient  to  the  set  (xv  .r2,  .r3),  then  the  (y)  polars  of  /  are 

(cf.  (61)) 

fvk  =  a£-*a*     (k  =  0,  1,  •••,  m).  (200) 

*  Clebscb,  Lindemann,  Vorlesungea  iiber  Geometrie. 

209 


210 


THE   THEORY   OF   INVARIANTS 


If  the  point  (</)  is  on  the  curve  /=  0,  the  equation  of  the 
tangent  at  (//)  is 

ata%-1  =  0.  (201) 

The  expression 


d.r. 


d 
dx0 


I  m  —  1  \n  —  1  \p  —  1    q 


hi 


\n  \p 


fyi   ^y-i 

dzx     dz2 


_d_ 

dxs 
dz0 


(202) 


is  sometimes  called  the  first  transvectant  of  /(#),  <K20>  ^(V), 
and  is  abbreviated  (/,  (f),  -v/r).      If 

/O)  =  a™  =  a'™  =  -..  4>(z)  =  bnx  =  b'?=  ...,  ^(*)  =  «Jf  =  #  =  - 

then,  as  is  easily  verified, 

(/,  </>,  y\r)  =  {abc)a"rlb'r{e"-x . 

This  is  the  Jacobian  of  the  three  forms.  The  rth  trans- 
vectant is 

(/,  0,  i/r)''  =  (abcya";-'b'J.-reP-r     (r  =  0,  1,  ...).      (203) 

For  r  =  2  and/=$  =  i/r  this  is  called  the  Hessian  curve. 
Thus 

( /,  /,/  ,2  _  ( abcyay-W-h-*-'2  =  0 

is  the  equation  of  the  Hessian.  It  was  proved  in  Chapter  I 
that  Jacobians  are  concomitants.  A  repetition  of  that  proof 
under  the  present  notation  shows  that  transvectants  are  like- 
wise concomitants.  In  fact  the  determinant  A  in  (202)  is 
itself  an  invariant  operator,  and 

A'  =(\pv)A. 

Illustration.  As  an  example  of  the  brevity  of  proof  which 
the  symbolical  notation  affords  for  some  theorems  we  may 
prove  that  the  Hessian  curve  of/  =  0  is  the  locus  of  all  points 
whose  polar  conies  are  degenerate  into  two  straight  lines. 


INVARIANTS   OF   TERNARY   FORMS 


211 


If  g  =  ax  =  /3§  =  •••  =  %M>-ri  +  •••  is  a  conic,  its  second  trans- 
vectant  is  its  discriminant,  and  equals 


(a/37)2  =  (2±ai/3273)2  =  6 


200 


a 


no 


a 


Ml 


^110       a020       a011 
al01       a011       a002 

,0  etc.     If  (a/87)2  =  0  the  conic  is  a 


since   a\  =  /3j  =  •••  =  a 
2-line. 

Now  the  polar  conic  of/ is 

P  =  a%a$-2  =  aX'"-2  =  '"•> 

and  the  second  transvectant  of  this  is 

(P,  P,  P)2  =  ( aa'a")2a™-2a'™-2a'Jm-2.  (204) 

But  this  is  the  Hessian  of  /  in  (v/)  variables.     Hence  if  ( g) 
is  on  the  Hessian  the  polar  conic  degenerates,  and  conversely. 
Every  symbolical  monomial  expression  <$>  consisting  of  fac- 
tors of  the  two  types  (abc),  ax  is  a  concomitant.     In  fact  if 

4>  =  (abcy(abdy  •••  axb%  •••, 


then 


4>'  = 


«A 

h 

Cx 

V 

«A 

h 

dK 

dy. 

K 

G* 

S 

K 

dp 

av 

K 

cv 

av 

K 

dv 

axb% 


since,  by  virtue  of  the  equations  of  transformation  a'x  =  ax,  •••. 
Hence  by  the  formula  for  the  product  of  two  determinants, 
or  by  (14),  we  have  at  once 

<f>'  =(\fjLv)p+q+-(abc~)p(abdy  •••  arxbx  •••  =  (\fivy+^  "<f). 

The  ternary  polar  of  the  product  of  two  ternary  forms  is 
given  by  the  same  formula  as  that  for  the  polar  of  a  product 
in  the  binary  case.  That  is,  formula  (77)  holds  when  the 
forms  and  operators  are  ternary. 

Thus,  the  formula  for  the  rth  transvectant  of  three  quali- 
ties, e.g. 

T=  (/,  <f>,  f)r  =  {abcYa'rrbrrcrr^ 


212  THE   THEORY   OF  INVARIANTS 

may  be  obtained  by  polarization  :  That  is,  by  a  process  analo- 
gous to  that  employed  in  the  standard  method  of  transvec- 
tion  in  the  binary  case.     Let 

(bc')1  =  b2c3-b3c2.  (bc)2  =  b3cx  -  bxc3,  (be)s  =  b^2  -  b2cy    (205) 

Then  am  =  (abc).  (206) 

Hence  T  may  be  obtained  by  polarizing  axl  r  times,  changing 
i/i  into  (£><?);  and  multiplying  the  result  by  bx~rcx-r.     Thus 


(a$bx,  cx,  dx~)2  =  - 


i)(i)^«A  +  (i)(o)a'5-r 

=  ^(acd)(bcd^)axcx  +  ^{acd)2bxcu 


Before  proceeding  to  further  illustrations  we  need  to  show 
that  there  exists  for  all  ternary  collineations  a  universal  co- 
variant.  It  will  follow  from  this  that  a  complete  fundamental 
system  for  a  single  ternary  form  is  in  reality  a  simultaneous 
system  of  the  form  itself  and  a  definite  universal  covariant. 
We  introduce  these  facts  in  the  next  paragraph. 

II.  Contragrediency.  Two  sets  of  variables  (xv  xv  £3), 
(uv  uv  w3)  are  said  to  be  contragredient  when  they  are  sub- 
ject to  the  following  schemes  of  transformation  respectively  : 

ajj  =  \xx[  +  ixxx'2  +  vxx'3 

V :    x2  =  \2x[  +  yu2.ro  +  p2x'3 

x3  =  \3x[  +  fiBa^2  +  v3x3 

U\  ~  ^l"l  +  ^2 "2  +  \sM3 
A  :     ?4  =  ^lWl  +  ^2?<2  +  ^3W3 

u'z  =  vlUl  +  v2u2  +  v3u3. 

Theorem.  A  necessary  and  sufficient  condition  in  order 
that  (x)  may  be  contragredient  to  (u)  is  that 

ut.=  ulx1  +  u2x2  +  u3x3 

shoidd  be  a  universal  covariant. 


INVARIANTS   OF  TERNARY   FORMS  213 

If  we  transform  ux  by  V  and  use  A  this  theorem  is  at  once 
evident. 

It  follows,  as  stated  above,  that  the  fundamental  system 
of  a  form  /  under  V,  A  is  a  simultaneous  system  of  /  and  ux 
(cf.  Chap.  VI,  §  4). 

The  reason  that  ur  =  u1x1  +  u2x2  does  not  figure  in  the  cor- 
responding way  in  the  binary  theory  is  that  cogrediency  is 
equivalent  to  contragrediency  in  the  binary  case  and  ux  is 
equivalent  to  (xy)  =  xxy%  —  x2yv  which  does  figure  very 
prominently  in  the  binary  theory.  To  show  that  cogredi- 
ency and  contragrediency  are  here  equivalent  we  may  solve 

u[  =  XjWj  +  \2w2 
u'2  =  ^ux  +  /*2m2, 
we  find 

—  O^O^i  =  X2w2  +  fi2(  —  m£), 

(\^)w2  =  \y2  +  /x:(-  u'j), 

which  proves  that  yx  =  +  uv  y2  =  —  ux  are  cogredient  to  xv 
xY     Then  ux  becomes  (j/:r)(cf.  Chap.  1,  §  3,  V). 

We  now  prove  the  principal  theorem  of  the  symbolic 
theory  which  shows  that  the  present  symbolical  notation 
is  sufficient  to  represent  completely  the  totality  of  ternary 
concomitants. 

III.  Theorem.  Every  invariant  formation  of  the  ordinary 
rational  integral  type,  of  a  ternary  quantic 

TT\  \m 

f=am> |fafaKa»'-rf2«4  (2n%  =  w)' 

can  be  represented  symbolically  by  three  types  of  factors,  viz. 

(ata),  (abti),  ax, 

together  with  the  universal  covariant  ux. 
We  first  prove  two  lemmas. 


214  THE   THEORY   OF   INVARIANTS 

Lemma  1.     The  following1  formula  is  true  : 


A"D"  = 


d 

3\j 

d 

d\2 

d 
d\3 

n 

\.2 

\3 

d 

d 
d/x2 

d 

H 

fJ-2 

H 

d 
dv2 

d 

dp3 

V\ 

V2 

V3 

=  <7,        (207) 


where  C  =f=  0  is  a  numerical  constant. 

In  proof  of  this  we  note  that  Dn,  expanded  by  the  multi- 
nomial theorem,  gives 

v  \h\h\h 

Oi^2  -/Vi)ts-     (Zij  =  n). 

Also    the    expansion   of    A"  is  given    by  the    same    formula 

where  now  (X^^t)  is  replaced  by  (- ).     We   may 

\d\r  dfxs  dvtJ 

call  the  term  given  by  a  definite  set  iv  iv  i3  of  the  exponents 

in  _D",  the  correspondent  of  the  term  given  by  the  same  set  of 

exponents  in  A".     Then,  in  A"_Z)n,  the  only  term  of  D"  which 

gives  a  non-zero  result  when  operated    upon   by  a  definite 

term  of  An  is  the  correspondent  of  that  definite  term.     But 

Dn  may  be  written 

ij  \h\h\h 

An  easy  differentiation  gives 

(j-£)  OM'O^o  >!s  h(h + h +H+1 )  (^)K^<>)!r !» 

and  two  corresponding  formulas  may  be  written  from  symme- 
try. These  formulas  hold  true  for  zero  exponents.  Employ- 
ing them  as  recursion  formulas  we  have  immediately  for 
A"Z)\ 


INVARIANTS   OF  TERNARY  FORMS  215 

A"D»  =  V  ( ,-7-j^— )\\h\h\h)2\h  +  h+  »8  +  l 

= ij  ciw)2iw+i= ki»)8(w + i>2(w + 2>-  (2°8) 

This  is    evidently  a   numerical    constant   (7=^0,  which  was 
to  be  proved  (cf.  (91)). 

Lemma  2.  If  P  is  a  product  of  m  factors  of  type  aA,  n  of 
type  /8M,  and  jo  of  type  7„,  then  AfcP  is  a  sum  of  a  number  of 
monomials,  each  monomial  of  which  contains  k  factors  of 
type  («/37),  m  —  k  factors  of  type  aK,  n  —  k  of  type  /8  ,  and 
p  —  k  of  type  7^. 

This  is  easily  proved.     Let  P  =  AB (7,  where 


Then 


a3P 


(>  =  7(i)7(2)  ...  7<p). 
^£  (7 


J)7i0 


r  =  l, 

8=  1, 


n 


Writing  down  the  six  such  terms  from  AP  and  taking  the 
sum  we  have 

Ai^C'^'V").^^  (209) 

r,s,t  "A    M/x    lv 

which   proves    the  lemma    for    k=  1,  inasmuch    as   —  has 

«r 

m  —  1  factors  ;  and  so  forth.  The  result  for  AkP  now  fol- 
lows by  induction,  by  operating  on  both  members  of  equation 
(209)  by  A,  and  noting  that  (a(r)/3(s)y(t))  is  a  constant  as  far 
as  operations  by  A  are  concerned. 

Let  us  now  represent  a  concomitant  of  /  by  <£(a,  #),  and 


216 


THE   THEORY   OF   INVARIANTS 


suppose  that  it  does  not  contain  the  variables  (w),  and  that 
the  corresponding  invariant  relation  is 

<£(a',  x\  -•^  =  (\fj,vy<]>(a,  x,  •••)•  (210) 

The  inverse  of  the  transformation  F"is 

x[  =  (X/xz/)-1  ICfiv)^  +  O^V2  +  OuOrfZv] 

etc.     Or,  if  we  consider  (z)  to  be  the  point  of  intersection  of 
two  lines 

v*=  i\xx  +  v2x2  +  y3.r3, 

wx  =  ivvi\  +  w2x2  +  it'Bxs, 
we  have 

x1:x2:x3  =  (yw\  :  (yw\  :  (w)3. 

Substitution  with  these  in  x'v  •••  and  rearrangement  of  the 
terms  gives  for  the  inverse  of  V 


r-i 


•To  = 


_  VplVv  —  VvWn 

(Xfiv) 

vvit\  —  VKWV 

(Xfiv) 

(Xixv) 

We  now  proceed  as  if  we  were  verifying  the  invariancy  of 
(/>,  substituting  from  V'1  for  x'v  x2,  x'3  on  the  left-hand  side  of 
(210),  and  replacing  a'm  m  b}r  its  symbolical  equivalent 
a^a™*a™3  (cf.  (199)).  Suppose  that  the  order  of  <f>  is  o>. 
Then  after  performing  these  substitutions  and  multiplying 
both  sides  of  (210)  by  (X/ii/)0'  we  have 

<K<laJNC»,  V*P>*  ~  *W,  •••)  =  CV"0**"  <£(a,  x,  ...), 

and  every  term  of  the  left-hand  member  of  this  must  contain 
w  +  (o  factors  with  each  suffix,  since  the  terms  of  the  right- 
hand  member  do.  Now  operate  on  both  sides  by  A.  Each 
term  of  the  result  on  the  left  contains  one  determinant  factor 
by  lemma   2,  and  in  addition    iv  +  w  —  1   factors    with   each 


INVARIANTS   OF   TERNARY  FORMS  217 

suffix.  There  will  be  three  types  of  these  determinant  fac- 
tors e.g. 

(afo),   (avw~)  =  ax,  (abv). 

The  first  two  of  these  are  of  the  form  required  by  the 
theorem.  The  determinant  (a5v)  must  have  resulted  by 
operating  A  upon  a  term  containing  axblxvv  and  evidently 
such  a  term  will  also  contain  the  factor  w^  or  else  W\.  Let 
the  term  in  question  be 

Then  the  left-hand  side  of  the  equation  must  also  contain 
the  term 

—  Rakby.vILwv, 

and  operation  of  A  upon  this  gives 

—  M^abw^v^, 
and  upon  the  sum  gives 

R^abvyw^  —  (abviyvy^. 

Now  the  first  identity  of  (212)  gives 

(abvywp  —  (abw")Vn  =  (bviv^a^.  —  {avw^b^  =  b^a^  —  b^dj.. 
Hence  the  sum  of  the  two  terms  under  consideration  is 

B>(J>x<*>*  —  &/»«*)» 

and  this  contains  in  addition  to  factors  with  a  suffix  /m  only 
factors  of  the  required  type  ar.  Thus  only  the  two  required 
types  of  symbolical  factors  occur  in  the  result  of  operating 
by  A. 

Suppose  now  that  we  operate  by  Aw+0i  upon  both  members 
of  the  invariant  equation.  The  result  upon  the  right-hand 
side  is  a  constant  times  the  concomitant  <£(«,  x)  by  lemma 
1.  On  the  left  there  will  be  no  terms  with  X,  /x,  v  suffixes, 
since  there  are  none  on  the  right.  Hence  by  dividing 
through  by  a  constant  we  have  (p(a,  x)  expressed  as  a  sum 
of  terms  each  of  which  consists  of  symbolical  factors  of  only 
two  types  viz. 

(abe),  «.,., 


218  THE   THEORY   OF   INVARIANTS 

which  was  to  be  proved.  Also  evidently  there  are  precisely 
o)  factors  ax  in  each  term,  and  w  of  type  (abc),  and  a>  =  0  if 
(f>  is  an  invariant. 

The  complete  theorem  now  follows  from  the  fact  that  any 
invariant  formation  of  /  is  a  simultaneous  concomitant  of  f 
and  ux.  That  is,  the  only  new  type  of  factor  which  can  be 
introduced  by  adjoining  ux  is  the  third  required  type  (abu). 

IV.  Reduction  identities.  We  now  give  a  set  of  identi- 
ties which  may  be  used  in  performing  reductions.  These 
ma}r  all  be  derived  from 

ax     bx     ( 


ay  Oy  Cy 

a,      b,      c, 


=  (abc)(xyz),  (211) 


as  a  fundamental  identity  (cf.  Chap.  Ill,  §  3,  II).  We  let 
uv  w2,  u3  be  the  coordinates  of  the  line  joining  the  points 
(x)  =  (xv  xv  xs),  O)  =  (yr  yv  j/g).     Then 

ux  :  u2  :  u3  =  (xy)x  :  (xy\  :  (xy\. 

Elementary  changes  in  (211)  give 

(bed)  a  x  —  (cdd)bx  +  ( dab)ex  —  (abc)dx  =  0, 
(bcu)ax  —  (cua)bj.  +  (uab)cr  —  (abc)ur  =  0,  (212) 

(abc)(def)  -  (dab)(cef)  +  (cda)(bef)  -  (bcd)(aef)  =  0. 

Also  we  have 

a,.bv  —  aj)r  =  (abu)i 

,     A    \  (213) 

Pawb  —  vbwa=  (abx). 

In  the  latter  case  (x)  is  the  intersection  of  the  lines  v,  w. 

To   illustrate    the    use    of    these    we    can    show    that    if 
/=  ar  =  •••  is  a  quadratic,  and  D  its  discriminant,  then 

(abc)(abd)crdx  =  ^  Df. 

In  fact,  by  squaring  the  first  identity  of  (212)  and  inter- 
changing the  symbols,  which  are  now  all  equivalent,  this 
result  follows  immediately  since  (abe)2  =  D. 


INVARIANTS   OF   TERNARY   FORMS  219 

SECTION   2.     TRANSVECTANT   SYSTEMS 

I.  Transvectants  from  polars.  We  now  develop  a  stand- 
ard transvection  process  for  ternary  forms. 

Theorem.     Every  monomial   ternary  concomitant   of  f= 

<f)  =  (abc)p(abd)q  •••  (bcd)r  •••  (abu)s(bcuy  •  ••  aa  •  ••, 

is  a  term  of  a  generalized  transvectant  obtained  by  polarization 
from  a  concomitant  <£x  of  lower  degree  than  cf>. 

Let  us  delete  from  cf>  the  factor  a%,  and  in  the  result 
change  a  into  v,  where  v  is  cogredient  to  u.  This  result 
will  contain  factors  of  the  three  types  (bcv),  (bed),  (buv), 
together  with  factors  of  type  br.  But  (uv)  is  cogredient  to 
x.  Hence  the  operation  of  changing  (uv)  into  x  is  invari- 
antive  and  (buv)  becomes  bx.  Next  change  v  into  u.  Then 
we  have  a  product  ^  of  three  and  only  three  types,  i.e. 

(6cw),  (bed),  bx, 
ff>1  =  (bcd)a  .-•  (bcuY  •••  byxc\  .... 

Now  <^>l  does  not  contain  the  symbol  a.  Hence  it  is  of 
lower  degree  than  cf>.  Let  the  order  of  </>  be  <o,  and  its  class 
fi.  Suppose  that  in  <f>  there  are  i  determinant  factors  con- 
taining both  a  and  w,  and  k  which  contain  a  but  not  u. 
Then 

<r  +  i  +  k  =  m. 

Also  the  order  of  <$>x  is 

(1,l==(o-{-2i  +  k—  m, 
and  its  class 

(Uj  =  fx  —  i  +  k. 

We  now  polarize  $>x  by  operating  [v— )  (#7-)  upon  it  and 
dividing  out  the  appropriate  constants.     If  in  the  resulting 


220  THE   THEORY   OF  INVARIANTS 

polar  we  substitute  v  =  a,  y=(au)  and  multiply  by  a^~i~k 
we  obtain  the  transvectant  (generalized) 

T  =  ((f)v  «',",  ?4)A'  \  (214) 

The  concomitant  <f>  is  a  term  of  t 

For  the  transvectant  t  thus  defined  k  +  i  is  called  the 
index.  In  any  ternary  concomitant  of  order  a>  and  class  /a 
the  number  a>  +  fi  is  called  the  grade. 

Definition.  The  mechanical  rule  by  which  one  obtains 
from  a  concomitant 

C  =  A.<X\xClo,x  '"  ar.ralua2u  "'  asui 

any  one  of  the  three  types  of  concomitants 

Cj  =  A^a^a^a^a^,.  •  ••  arraUta2u  •••  asu, 
C2  =  Aalaia2r  •••  o>rx(h,u(hu  '"  asw> 
C3  =  A^a^a^a^  •  ••  a,,,.«4tt  ...  asil, 

is  called  convolution.     In  this  aia,  indicates  the  expression 

^ll^ll  ~^~  ai2a12  "I"  al3CC13" 

Note  the  possibility  that  one  a  might  be  a-,  or  one  a  might 
be  u. 

II.  Theorem.  7%e  difference  between  any  two  terms  of  a 
transvectant  t  equals  reducible  terms  whose  factors  are  concom- 
itants of  lower  grade  than  t,  plus  a  sum  of  terms  each  term 
of  which  is  a  term  of  a  transvectant  r  of  index  <  k  +  t, 

In  this,  <f)l  is  of  lower  grade  than  (f>x  and  is  obtainable  from  the 
latter  by  convolution. 

Let  ^>j  be  the  concomitant  0  above,  where  A  involves 
neither  u  nor  x.     Then,  with  X  numerical,  we  have  the  polar 

=  A1alya2y  ...  aivai+lx  •  ••  arJ.alv  ...  akvak+lu  •••  asu.  (215) 


INVARIANTS   OF   TERNARY  FORMS  221 

Now  in  the  ith.  polar  of  a  simple  product  like 

two  terms  are  said  to  be  adjacent  when  they  differ  only  in 
that  one  has  a  factor  of  type  7^7^  whereas  in  the  other  this 
factor  is  replaced  by  7/,/y,,,-  Consider  two  terms,  tv  t2  of  P. 
Suppose  that  these  differ  only  in  that  avvaKUaflyajx  in  tx  is  re- 
placed in  t2  by  ar,u^KV<ihxajij  •     Then  £j  —  t2  is  of  the  form 

^i  —  t2  =  Bya^a^a^aj^.  —  ami^Kvahxajy)' 

We  now  add  and  subtract  a  term  and  obtain 

h-h  =  Blam'a<ii(.ah,Aj.r-ahraJy)  +  ahxajy(anvuKU  -  ctmaltl)'] .    (216) 

Each  parenthesis  in  (216)  represents  the  difference  between 
two  adjacent  terms  of  a  polar  of  a  simple  product,  and  we 
have  by  (213) 

tx  -  t2  =  B^yx^a^a^a^  +  JS(aKaI)(wv))a/(J.a,> .     (217) 

The  corresponding  terms  in  t  are  obtained  by  the  replace- 
ments v  =  «,  y  =  («m).     They  are  the  terms  of 

8=  —  i?'((aw)(a;io,):c)aai)ocKM  —  JB'((aM)aKan)(aJaw)aAx, 

or,  since 

{(au)(ahaj*)x)  =  (aaha^)ux  —  (ahaJu')ax, 
of 

S  =  B' XaAa/w)<V<W*.r  -  ^'(%«/a)«WW*r 

+  B'{anaK(au))(Kajau')ahx, 

where  B  becomes  B'  under  the  replacements  v  =  a,  y  ={au). 
The  middle  term  of  this  form  of  S  is  evidently  reducible, 
and  each  factor  is  of  lower  grade  than  t.  By  the  method 
given  under  Theorem  I  the  first  and  last  terms  of  S  are  re- 
spectively terms  of  the  transvectants 

r1=(B1(ahaju')amiaKU,  a™,  w^1)*'*-1* 
t2  =  (B^a  avx)ajxahx,  a?,  <+1)*_1'i+1- 
The  middle  term  is  a  term  of 

t3  =(-  B^a^u^a^  a™,  MJT1)*+1,i-1  •  ux. 


222  THE   THEORY   OF   INVARIANTS 

In  each  of  these  B1  is  what  B  becomes  when  v  =  u,  y  =  x ; 
and  the  first  form  in  each  transvectant  is  evidently  obtained 
from  u ,(f)1  =  Ouxhy  convolution.  Also  each  is  of  lower  grade 
than  <j>v 

Again  if  the  terms  in  the  parentheses  in  form  (216)  of 
any  difference  tx  —  t2  are  not  adjacent,  we  can  by  adding  and 
subtracting  terms  reduce  these  parentheses  each  to  the  form  * 

[01-t2)  +  (t2-t3)+  ...(t^-t/)],  (218) 

where  every  difference  is  a  difference  between  adjacent  terms, 
of  a  simple  polar.  Applying  the  results  above  to  these  dif- 
ferences T,-  —  Ti+1  the  complete  theorem  follows. 

As  a  corollary  it  follows  that  the  difference  between  the 
whole  transvectant  t  and  any  one  of  its  terms  equals  a  sum 
of  terms  each  of  which  is  a  term  of  a  transvectant  of  a"1  with 
a  form  ^  of  lower  grade  than  <j>v  obtained  by  convolution 
from  the  latter.     For  if 

T  =  I^Tj  -f  V2T2  +    •••    +  VTTT  +    ••• 

where  the  vs  are  numerical,  then  rr  is  a  term  of  t.  Also 
since  our  transvectant  t  is  obtained  by  polarization,  1v{  =  1. 
Hence 

T  -  Tr  =  Vl(jl  -  Tr)  +  V2(T2  -  Tr)  +    •••, 

and  each  parenthesis  is  a  difference  between  two  terms  of  t. 
The  corollary  is  therefore  proved. 

Since  the  power  of  ux  entering  t  is  determinate  from  the 
indices  k,  i  we  may  write  r  in  the  shorter  form 

The  theorem  and  corollary  just  proved  furnish  a  method 
of  deriving  the  fundamental  system  of  invariant  formations 
of  a  single  form/=  a%  by  passing  from  the  full  set  of  a  given 
degree  *  —  1,  assumed  known,  to  all  those  of  the  fundamental 

*Isserlis.  On  the  ordering  of  terms  of  polars  etc.  Proc  London  Math. 
Socielp.  ser.  2,  Vol.6  (1908). 


INVARIANTS   OF   TERNARY    FORMS  223 

system,  of  degree  i.  For  suppose  that  all  of  those  members 
of  the  fundamental  system  of  degrees  <  i  —  1  have  been 
previously  determined.  Then  by  forming  products  of  their 
powers  we  can  build  all  invariant  formations  of  degree  i  —  1. 
Let  the  latter  be  arranged  in  an  ordered  succession 

in  order  of  ascending  grade.  Form  the  trans vectants  of 
these  with  a™,  t?-=(<£0),  a™)k,i.  If  ri  contains  a  single  term 
which  is  reducible  in  terms  of  forms  of  lower  degree  or  in 
terms  of  transvectants  rjtj'  <j,  then  t;-  may,  by  the  theorem 
and  corollary,  be  neglected  in  constructing  the  members  of 
the  fundamental  system  of  degree  i.  That  is,  in  this  con- 
struction we  need  only  retain  one  term  from  each  trans- 
vectant  which  contains  no  reducible  terms.  This  process  of 
constructing  a  fundamental  system  by  passing  from  degree 
to  degree  is  tedious  for  all  systems  excepting  that  for  a 
single  ternary  quadratic  form.  A  method  which  is  equiva- 
lent but  makes  no  use  of  the  transvectant  operation  above 
described,  and  the  resulting  simplifications,  has  been  applied 
by  Gordan  in  the  derivation  of  the  fundamental  system  of  a 
ternary  cubic  form.  The  method  of  Gordan  was  also  suc- 
cessfully applied  by  Baker  to  the  system  of  two  and  of  three 
conies.  We  give  below  a  derivation  of  the  system  for  a 
single  conic  and  a  summary  of  Gordan's  system  for  a  ternary 
cubic  (Table  VII). 

III.    Fundamental  systems  for  ternary  quadratic  and  cubic. 

Let/=  drx  =  6|  =  •••  .  The  only  form  of  degree  one  is  /  it- 
self.    It  leads  to  the  transvectants 

(a%  £2)0.1  =  (abu)axbt  =  0,  (a2,  J2)0'2  =  («6m)2=  L. 

Thus  the  only  irreducible  formation  of  degree  2  is  L.  The 
totality  of  degree  2  is,  in  ascending  order  as  to  grade, 


224  THE   THEORY  OF  INVARIANTS 

All  terms  of  (/2,  /)*■ *  are  evidently  reducible,  i.e.  contain 
terms  reducible  by  means  of  powers  of /and  L.     Also 

{(abuf,  cl)1'°=  (abc)(abu)cx 

=  ^(abc^\_(abu)cx  +  (beu)ax  +  (caii)br~\  =  ^(abcyux, 

((abu)2,  ciy-»=(abcy=D. 

Hence  the  only  irreducible  formation  of  the  third  degree  is 
D.  Passing"  to  degree  four,  we  need  only  consider  trans- 
vectants  of  fL  with/.  Moreover  the  only  possibility  for  an 
irreducible  case  is  evidently 

(/L,/)1'1=  (abd^(abu)<lcdu)cx 
=  ^(abu)Qcdu^\_(abd)cx  4-  (bcd)ax  +  (dca)bx  -f  {acb~)dx~\  =  0. 

All  transvectants  of  degree  >  4  are  therefore  of  the  form 

CW/)*«(t'  +  *<8), 

and  hence  are    reducible.      Thus    the   fundamental    system 

off  is 

wx,  /,  Z,  D. 

The  explicit  form  of  D  was  given  in  §  1.  A  symmetrical 
form  of  L  in  terms  of  the  actual  coefficients  of  the  conic  is 
the  bordered  discriminant 

*200       all0       al01       Ul 

hl<)       rt020       a011       U2 

Z101       a011       a002       U3 
itj  W2  Mg  0 

To  verify  that  i  equals  this  determinant  we  may  expand 
(a5w)2  and  express  the  symbols  in  terms  of  the  coefficients. 

We  next  give  a  table  showing  Gordan's  fundamental 
system  for  the  ternary  cubic.  There  are  thirty-four  in- 
dividuals in  this  system.  In  the  table,  i  indicates  the 
degree. 

The    reader   will   find    it    instructive    to   derive   by   the 


INVARIANTS  OF  TERNARY  FORMS 


225 


methods  just  shown  in  the  case  of  the  quadratic,  the  forms 
in  this  table  of  the  first  three  or  four  desrrees. 


TABLE    VII 


i 

Invariant  Formation 

0 

ux 

1 

< 

2 

(abu)2axbx 

3 

{abu)2{bcu)axc%  ax=:(abc)2axbxcx,  si  —  (abc)  (abu)  (acu)  (bcu) 

4 

{aau)a2xa%  a,sla'x,  S  =  a3s,  pft  =  (abti)2(cdu)2(bcu)(adu) 

5 

assl(abu)axbl,  asbssua2xb2x,  as(abu)2slbx,  t%  —  asbssu(abn)2 

6 

asbssu(bcii)a2bxc2,  ass2(abu)2(bcu)c2,  att2ta2x,  T  =  a\ 

7 

slpl(spx),  attl(abu)axb%  atbttua%b2,  attl(abu)2bx 

8 

atbttu(bcu)a2xbxcx,  qi  =  atbtcta2xb2cl,  attl(abu)2(bcu)c2,  s2t2(stx) 

9 

(aqu)axqx,  p5ut2u{ptx),  ats%tua%(stx) 

10 

atbtslalb2x(stx),  ats2utu(abu)2bx(stx) 

11 

(aqu)aW: 

12 

(aaq)alalqx;,  2>£s^(jps£) 

IV.    Fundamental  system  of  two  ternary  quadrics.     We 

shall  next  define  a  ternary  transvectant  operation  which 
will  include  as  special  cases  all  of  the  operations  of  trans- 
vection  which  have  been  employed  in  this  chapter.  It  will 
have  been  observed  that  a  large  class  of  the  invariant  for- 
mations of  ternary  qualities,  namely  the  mixed  concomitants, 
involve  both  the  (z)  and  the  (u)  variables.  We  now  assume, 
quite  arbitrarily,  two  forms  involving  both  sets  of  variables 
e.g. 

<j>  =  Aau.a2j.  •••  a,,r«iua2„  •••  «w, 


226  THE   THEORY   OF   INVARIANTS 

in  which  A,  B  are  free  from  (x)  and  (w).  A  transvectant 
of  </>,  and  -v/r  of  four  indices,  the  most  general  possible,  may 
be  delined  as  follows:   Polarize  (f)  by  the  following  operator, 

wherein  e,-,  t;-,  -cr,,  v{  =  0  or  1,  and 

2e  =  i,  St  =  j\  So-  =  &,   2v  =  ? ;   i  +  j  ^  r,  k  -+-  /  ^  s. 
Substitute  in  the  resulting  polar 

(a)  3#>=/8p.  (i?  =  l,  2,  .-,  0, 
(*)  V?  =  (J>J*)  (^  =  1,2,  ...,./), 
(0  «<»-ft,  Q>  =  1,  2,  ...,*), 

(d)  Vf=(^)  (^  =  1,2,  ...,?), 

and  multiply  each  term  of  the  result  by  the  bx,  yS„  factors  not 
affected  in    it.     The   resulting   concomitant  t   we   call  the 

transvectant  of  </>  and  -v/r  of  index  (  t,   A  and  write 
An  example  is 

+  fl^«{!("i')l")+":y(i1("lJ2"')- 

If,  now,  we  introduce  in  place  of  $  successively  products  of 
forms  of  the  fundamental  system  of  a  conic,  i.e.  of 

/=  a%  L  =  «2  =  (  a'a"u)2,  D  =  (aaV)2, 

and  for  -\|r  products  of  forms  of  the  fundamental  system  of  a 
second  conic, 

g  =  b%  L<  =  £2  =  (b'b"u)\  I)'  =  (bb'h")\ 

we  will  obtain  all  concomitants  of /and  g.  The  fundamental 
simultaneous  system  of/,  #  will  be  included  in  the  set  of 


INVARIANTS  OF  TERNARY  FORMS 


2'2~ 


trans vectants  which  contain  no  reducible  terms,  and  these 
we  may  readily  select  by  inspection.  They  are  17  in  num- 
ber and  are  as  follows  : 

3>  =OI,  i!)8;  g  =  (ahu)\ 
<7j  =  (a%  5^)0,  x  _  (abu)a,.br, 

122  —  \~&  A-Wo.O  —     P"1 

o2  =  01,  ^)o',o  =  afiaA, 

03  =  («2,  62)0,  o  =  abaubx, 

-"112  =  \atti  bx)%  o  =  abi 

Cb  =  («£,  62/32)o;  o  =  ab(apx)bxl2u, 

^6  =  («&  ^Doio  =  ^WJA. 

(77  =  (a2«2,  J|)0,i  =  «6(a6w)^.«tt, 
6r8  =  (a2.<4  £2)1,  o  =  a^(a/3a;)a;.aM, 
#  =  (/£,  ^01;  i  =  apab(a@x)axbx, 
r  =  (/£,  ^')1;  o  =  a^b(abu)aur3w 
Kx  =  (fL,  gL'  )i; }  =  dfi(abu)au(a/3x)bx, 
K2  -  (/-£>  ^02;  1  =  ab(abu)l3u(afix}ax, 
K3  =  (/L,  #i7)i; J  =  afiab(abu)(afix). 

The  last  three  of  these  are  evidently  reducible  by  the  simple 
identity 

(abu)(afix')  =  \a^     bp     /3U 

|  a,     6r     mx 


228  THE   THEORY   OF   INVARIANTS 

The  remaining  14  are  irreducible.  Thus  the  fundamental 
system  for  two  ternary  quadrics  consists  of  20  forms.  They 
are,  four  invariants  i),  D',  Auv  A122 ;  four  co variants  /,  g, 
F,  G-;  four  contravariants  Z,  L\  4>,  F;  eight  mixed  con- 
comitants Qi(i  =  \,  •••,  8). 

SECTION  3.     CLEBSCH'S   TRANSLATION   PRINCIPLE 

Suppose  that  (?/),  (z)  are  any  two  points  on  an  arbitrary 
line  which  intersects  the  curve /=  a™  =  0.     Then 

ux  :  w2  :  u3  =  Q/z  )j  :  (yz\  :  (yz\ 

are  contragredient  to  the  ar's.  If  (x~)  is  an  arbitrary  point 
on  the  line  we  may  write 

x\  =  7?i#i  +  7hzv  -r2  =  VitJz  +  ltfv  »3  =  V1I/3  +  VoJv 

and  then  (_rjv  ?;2)  may  be  regarded  as  the  coordinates  of  a 
representative  point  (x)  on  the  line  with  (y),  (z)  as  the  two 
reference  points.     Then  a,,  becomes 

and  the  (?;)  coordinates  of  the  m  points  in  which  the  line 
intersects  the  curve  /=  0  are  the  in  roots  of 

9  =  9"  =  OVh  +  azih)m=  ( byVl  +  hzih)m=  .... 

Now  this  is  a  binary  form  in  symbolical  notation,  and  the 
notation  differs  from  the  notation  of  a  binary  form  h  =  a'" 
=  (a1x1  -\-  a2x^)m  =  ...  only  in  this,  that  av  a2  are  replaced  by 
a„,  az  respectively.      Any  invariant, 

Ix=  1k(aby(acy  •-., 

of  h  has  corresponding  to  it  an  invariant  i"of  g, 

1=  1h(aybz  —  azby)p(«yt\  -  «zeyy  ••-. 


INVARIANTS   OF  TERNARY  FORMS  229 

If  /=  0  then  the  line  cuts  the  curve /=  «'"  =  0  in  m  points 
which  have  the  projective  property  given  by  Ix  —  0.  But 
(cf.  (213)), 

(aybz  —  a2by)  =  (abu'). 
Hence, 

Theorem.  If  in  any  invariant  I1  =  *2k(ab)p(ac)q  •••  of  a 
binary  form  h  =  a"1  =  (^alx1  +  «2^2)m ■=  •••  we  replace  each  second 
order  determinant  (a&)  by  the  third  order  determinant  (a6w), 
and  so  o?i,  the  resulting  line  equation  represents  the  envelope  of 
the  line  uv  when  it  moves  so  as  to  intersect  the  curve  f =  a'"  = 
(a^  +  a%x2  +  ^zxz)m  —  0  *w  m  points  having  the  projective 
property  Ix  =  0. 

By  making  the  corresponding  changes  in  the  symbolical 
form  of  a  simultaneous  invariant  I  of  any  number  of  binary 
forms  we  obtain  the  envelope  of  ux  when  the  latter  moves  so 
as  to  cut  the  corresponding  number  of  curves  in  a  point 
range  which  constantly  possesses  the  projective  property 
/=  0.  Also  this  translation  principle  is  applicable  in  the 
same  way  to  co variants  of  the  binary  forms. 

For  illustration  the  discriminant  of  a  binary  quadratic 
h  =  a2  =  b2  =  •••  is  D=(ab~)%.  Hence  the  line  equation  of 
the  conic /=  a|  =  (a^x^  +  a2x2  +  a3x3)2  =  •••  =  0  is 

L=(abu)2=0. 

For  this  is  the  envelope  of  ux  when  the  latter  moves  so  as  to 

touch  /=  0,  i.e.  so  that  D=  0  for  the  range  in  which  ux  cuts 

/=0. 

The    discriminant  of   the   binary  cubic    h  =  (a1x1  +  a2x2)s 

=  J1=  •••  is 

B  =  (aby(ac~)(bd)(cdy. 

Hence  the  line  equation  of  the  general  cubic  curve /= 
4=  ...  is  (cf.  Table  VII) 

ptiu  =  L=  (abu)2(acu)(bdu}(cdu')2=  0. 


230  THE   THEORY   OF   INVARIANTS 

We  have  shown  in  Chapter  I  that  the  degree  i  of  the  dis- 
criminant of  a  binary  form  of  order  m  is  2(ra  —  1).  Hence 
its  index,  and  so  the  number  of  symbolical  determinants  of 
type  (a£>)  in  each  term  of  its  symbolical  representation,  is 

k  =  \  im  =  m(m  —  1). 

It  follows  immediately  that  the  degree  of  the  line  equation, 
i.e.  the  class  of  a  plane  curve  of  order  m  is,  in  general, 
m(m  —  1). 

Two  binary  forms  hx  =  a™  =  a!™  =  •••,  h2  =  b'"  =  •••,  of  the 
same  order  have  the  bilinear  invariant 

I=(ab)m. 

If  1=  0  the  forms  are  said  to  be  apolar  (cf.  Chap.  Ill, 
(71));  in  the  case  m  =  2,  harmonic.  Hence  (abu)m  =  0  is 
the  envelope  of  ux  =  0  when  the  latter  moves  so  as  to  inter- 
sect two  curves /=  a™  =  0,  g  =  b™  =  0,  in  apolar  point  ranges. 


APPENDIX 

EXERCISES   AND   THEOREMS 

1.  Verify  that  7=  a0a4  —  4aja3  +3  a!  is  an  invariant   of   the 
binary  quartic 

/  =  a^x\  +  4  0^X2  +  6  a^fa-f  +  4  a^xl  +  u4£2> 

for  which  /'=(X^Z 

2.  Show  the  invariancy  of 

«i(«o#i  +  a^)  —  (^(a!^  +  a2x2), 
for  the  simultaneous  transformation  of  the  forms 
/=  a0xL  +  ayx2, 
g=a0aZ  +  2  a^x^  +  a.x\. 

Give  also  a  verification  for  the  covariant  C  of  Chap.  I,  §  1,  V, 
and  for  J^  ±  of  Chap.  II,  §  3. 

3.  Compute  the  Hessian  of  the  binary  quintic  form 

/=  cttfxl  +  5  a^x.,  +  •  •  •. 
The  result  is 

\  H  =  (a0a2  —  af)xl  +  3(a0a3  —  ava2)xlx2  +  3(a0a4  +  a^  —  2  al)^^ 
-f-(a0a5  +  7  c^  —  8  o^a^a^el  +  3(0^5  +  a2a\  —  2  af)^! 
+3(a2a5  —  a3tt4)«1^  +(a3a5  —  a\)x\. 

4.  Prove  that  the  infinitesimal  transformation  of  3-space  which 
leaves  the  differential  element, 

(j  =  dx1  +  dyj  +  dz2, 

invariant,  is  an  infinitesimal  twist  or  screw  motion  around  a 
determinate  invariant  line  in  space.  (A  solution  of  this  problem 
is  given  in  Lie's  Geometrie  der  Beiiihrungstransformationen. 
§  3,  p.  206.) 

231 


232  THE   THEORY  OF  INVARIANTS 

5.  The  function 

q  =  a20a2  +  a0o|  +  a0a°i  +  afa2  —  a§  —  a% 
is  a  formal  invariant  modulo  3  of  the  binary  quadratic 

/=  c/oXj  +  2  a^Xo  +  a^of  (Dickson). 

6.  The  function  a0a3  +  axa2  is  a  formal  invariant  modulo  2  of 
the  binary  cubic  form. 

7.  Prove  that  a  necessary  and  sufficient  condition  in  order  that 
a  binary  form  /  of  order  m  may  be  the  mth  power  of  a  linear 
form  is  that  the  Hessian  covariant  of  /  should  vanish  identically. 

8.  Show  that  the  set  of  conditions  obtained  by  equating  to 
zero  the  2  m  —  3  coefficients  of  the  Hessian  of  exercise  7  is  re- 
dundant, and  that  only  m  —  1  of  these  conditions  are  independent. 

9.  Prove  that  the  discriminant  of  the  product  of  two  binary 
forms  equals  the  product  of  their  discriminants  times  the  square 
of  their  resultant. 

10.  Assuming  (y)  not  cogredient  to  (x),  show  that  the  bilinear 
form 

/=  %aikx{yk  =  ctu^i  +  a12x:y2  +  (hv^i  +  <h&&2, 

has  an  invariant  under  the  transformations 

a?i  =  «,&  +  &&,  *2  =  yi£  +  8&, 

Vi  =  ChVi  +  fan  !h  =  7iV\  +  ^ 
in  the  extended  sense  indicated  by  the  invariant  relation 


aii    «2i 

«12        a22 

= 

«1        Pi 

yi    <$! 

«11        «21 

11.  Verify  the  invariancy  of  the  bilinear  expression 

Hfg  =  «u&22  +  «22&u  —  «iAi  -  <hfim 
for  the  transformation  by  r  of  the  two  bilinear  forms 
/  =  ^aikx{yk,  g  =  S&jjftjfe 

12.  As  the  most  general  empirical  definition  of  a  concomitant 
of  a  single  binary  form  /  we  may  enunciate  the  following  :  Any 
rational,  integral  function  <j>  of  the  coefficients  and  variables  of/ 


APPENDIX  233 

which  needs,  at  most,  to  be  multiplied  by  a  function  ^  of  the 
coefficients  in  the  transformations  T,  in  order  to  be  made  equal 
to  the  same  function  of  the  coefficients  and  variables  of  /',  is  a 
concomitant  of/. 

Show  in  the  case  where  <f>  is  homogeneous  that  ^  must  reduce 
to  a  power  of  the  modulus,  and  hence  the  above  definition  is 
equivalent  to  the  one  of  Chap.  I,  §  2.  (A  proof  of  this  theorem 
is  given  in  Grace  and  Young,  Algebra  of  Invariants,  Chapter  II.) 

13.  Prove  by  means  of  a  particular  case  of  the  general  linear 
transformation  on  p  variables  that  any  p-ary  form  of  order  ra, 
whose  term  in  ccf  is  lacking,  can  always  have  this  term  restored 
by  a  suitably  chosen  linear  transformation. 

14.  An  invariant  cj>  of  a  set  of  binary  quantics 

/i  =  «o«T  H ,  U  =  Mi  +  •"»  /s  =  V?  +  ••  •, 

satisfies  the  differential  equations 

m*  =(a0~+ 2  aJ-  +"■  +  ™«,»-.r  +  h~  +  2&i^ 

\    octy  oa2  oam         oOj  do2 

+  -  +  c0A+  ...V  =  o, 

o<h  J 

20<£  =(mai—-  +  O  -  l)<h— -  +  •••  +  am- ■  +  nbl—- 

+  (n  -  1)62J-  +  ».  +&J-  +  -V  =  0. 

3&i  c)c0       y 

The  covariants  of  the  set  satisfy 

SO  -  a?2—  )4>  =  0, 

dxj 

SO-a^Wo. 
5ic2y 

15.  Verify  the  fact  of  annihilation  of  the  invariant 
J=  6 


«0        «1        «2 

ax     a2     a3 
a2     «3     a4 


of  the  binary  quartic,  by  the  operators  O  and  O. 


234  THE   THEORY  OF  INVARIANTS 

16.  Prove  by  the  annihilators  that  every  invariant  of  degree  3 
of  the  binary  quartie  is  a  constant  times  J. 

(Suggestion.  Assume  the  invariant  with  literal  coefficients  and  operate 
by  fi  and  O. ) 

17.  Show  that  the  covariant  J^  A  of  Chap.  II,  §  3  is  annihilated 
by  the  operators 

SQ-afcA,  so-** 

axx  ox2 

18.  Find  an  invariant  of  respective  partial  degrees  1  and  2,  in 
the  coefficients  of  a  binary  quadratic  and  a  binary  cubic. 

The  result  is 

7=  OoC&^s  -  &!)—  a1(6„63  -  6i&2)+a2(&0&2  -  bf). 

19.  Determine  the  index  of  /in  the  preceding  exercise.  State 
the  circumstances  concerning  the  symmetry  of  a  simultaneous 
invariant. 

20.  No  covariant  of  degree  2  has  a  leading  coefficient  of  odd 
weight. 

21.  Find  the  third  polar  of  the  product  f  •  g,  where  /  is  a 
binary  quadratic  and  g  is  a  cubic. 

The  result  is 

Uti)+  =  Tv(f9y3  +  §fV9Vi  +  Sfytiv)- 

22.  Compute  the  fourth  transvectant  of  the  binary  quintic  / 
with  itself. 

The  result  is 

(ft  f)*  =  2(«o«4  —  4  a{a3  +  3  af)xl  +  2(a0ab  -  3  axaA  +  2  a^x^ 
+2(a1afi  —  4  a2a4  +  3  a§)«|. 

23.  If  F=<rr!/icI,  prow 


APPENDIX  235 

24.  Express  the  covariant 

Q=(ab)\cb)ciax 

of  the  binary  cubic  in  terms  of  the  coefficients  of  the  cubic  by  ex- 
panding the  symbolical  Q  and  expressing  the  symbol  combina- 
tions in  terms  of  the  actual  coefficients.     (Cf.  Table  I.) 

25.  Express  the  covariant  +j  =  ((/,/) \  ff  of  a  binary  quintic 
in  terms  of  the  symbols. 

The  result  is 

_  j  =(aby(bcy-(eayaxbxcx  =  -  {ab)Xac)(bc)^x. 

26.  Let  <£  be  any  symbolical  concomitant  of  a  single  form  /, 
of  degree  i  in  the  coefficients  and  therefore  involving  i  equivalent 
symbols.  To  fix  ideas,  let  <f>  be  a  monomial.  Suppose  that  the 
i  symbols  are  temporarily  assumed  non-equivalent.  Then  <f>, 
when  expressed  in  terms  of  the  coefficients,  will  become  a  simul- 
taneous concomitant  fa  of  i  forms  of  the  same  degree  as  /,  e.g. 

f=  a0x™  -f-  mc^xl'^x.,  +  •••, 
/i  =  b0xy  +  mb^-^Xz  +  -.., 


/»_i  =  l<p%  +  m,lvv';'  %  + 


Also  <^  will  be  linear  in  the  coefficients  of  each  /,  and  will  reduce 
to  c/>  again  when  we  set  bi  =  ■■■  =  l-  =  a,-,  that  is,  when  the  symbols 
are  again  made  equivalent.  Let  us  consider  the  result  of  operat- 
ing with 

oaQ  oax  oam     \    oaj 

upon  fa  This  will  equal  the  result  of  operating  upon  fa,  the 
equivalent  of  8,  and  then  making  the  changes 

6.  =  ...  =  I.  =  as  (j  =  0,  •»,  m). 

Now  owing  to  the  law  for  differentiating  a   product  the  result 

of  operating  — -  upon  <f>  is  the  same  as  operating 

±+±+...+± 

Sa,     db.  W, 


236  THE   THEORY   OF   INVARIANTS 

upon  fa  and  then  making  the  changes  b  =  •••  =  I  =  a.  Hence  the 
operator  which  is  equivalent  to  8  in  the  above  sense  is 

*-(*fi)+W>-+te 

When  8j  is  operated  upon  fa  it  produces  i  concomitants  the  first  of 
which  is  fa  with  the  a's  replaced  by  the  />'s,  the  second  is  fa  with 
the  6's  replaced  by  the  />'s,  and  so  on.     It  follows  that  if  we  write 

<  =  Pv>%  4-  mpiO^^Xi  +  •••, 

and 

4>=(aby(ac)'.:a<>M-~, 

we  have  for  8<£  the  sum  of  i  symbolical  concomitants  in  the  first 
of  which  the  symbol  a  is  replaced  by  w,  in  the  second  the  symbol 
b  by  7r  and  so  forth. 

For  illustration  if  <f>  is  the  covariant  Q  of  the  cubic, 

Q  =  (ab)\cb)clax, 
then 

8Q  =(tt&)2(c6)c>,  +(air)\cTr)c\ax  +  (a6)2(7r&)ir2ax. 

Again  the  operator  8  and  the  transvectant  operator  O  are 
evidently  permutable.  Let  g,  h  be  two  covariants  of /and  show 
from  this  fact  that 

%,  hy=(Bg,  hy+(g,8hy. 

27.   Assume 

a  =  (/,/)2  =  W«A  =  4 

Q  =  (/,  (/,  /)2)  =  (<*K*»  =  (a&)*(c&)c»aa  =  % 

B  =  (A,  A)2  =  (aby(cdy(ac)(bd), 
and  write 

Then  from  the  results  in  the  last  paragraph  (26)  and  those  in 
Table  I  of  Chapter  III,  prove  the  following  for  the  Aronhold 

polar  operator  8  =(  Q—  ] : 


APPENDIX  237 

¥=Q, 

SA  =  2(aQ)2ax&  =  2(/,  Q)2  =  0, 
*Q  =  2(/,  (/,  QY)  +(Q,  A)=  -  1 72/, 
Si?  =  4(A,  (/  Q)2)2  =  0. 

28.  Demonstrate  by  means  of  Hermite's  reciprocity  theorem 
that  there  is  a  single  invariant  or  no  invariant  of  degree  3  of  a 
binary  quantic  of  order  ra  according  as  ra  is  or  is  not  a  multiple 
of  4  (Cayley). 

29.  If  /is  a  quartic,  prove  by  Gordan's  series  that  the  Hessian 
of  the  Hessian  of  the  Hessian  is  reducible  as  follows : 

(OH,  H)\  (H,  Hyy  =  -  ^vjf+ 1//^-^-3). 

Adduce   general  conclusions  concerning  the  reducibility  of  the 
Hessian  of  the  Hessian  of  a  form  of  order  m. 

30.  Prove  by  Gordan's  series, 

(a»)!j)2={^Tv(^n 

where  i  =  (/,  /)4,  and  /  is  a  sextic.     Deduce  corresponding  facts 
for  other  values  of  the  order  ra. 

31.  If /is  the  binary  quartic 

/=<**  =  &*=  c*=..., 

show  by  means  of  the  elementary  symbolical  identities  alone  that 
(ab)\acfbyx  =  J/,  (aby. 
(Suggestion.     Square  the  identity 

2(ab)(ac)bzcz  =(ab)2c2x  +{ac)%l  -(6c)2a|.) 

32.  Derive  the  fundamental  system  of  concomitants  of  the 
canonical  quartic 

X4+F4+6raX2r2, 

by  particularizing  the  a  coefficients  in  Table  II. 

33.  Derive  the  syzygy  of  the  concomitants  of  a  quartic  by 
means  of  the  canonical  form  and  its  invariants  and  covariants. 


238  THE   THEORY  OF  INVARIANTS 

34.  Obtain  the  typical  representation  and  the  associated  forms 
of  a  binary  quartic,  and  derive  by  means  of  these  the  syzygy  for 
the  quartic. 

The  result  for  the  typical  representation  is 

f  '/(y)  =?  +  3 Hfrf  +  4  Ttr?  +  Q  if*  -  | IP)rf. 
To  find  the  syzygy,  employ  the  invariant  J. 

35.  Demonstrate  that  the  Jacobian  of  three  ternary  forms  of 
order  m  is  a  combinant. 

36.  Prove  with  the  aid  of  exercise  26  above  that 

(/,  </,)*+*  =(aa)'r+1a:-°-^a'r2r~1 
is  a  combinant  of /  =  a"  and  <£  =  a". 

37.  Prove  that  Q  =(ab)(bc)(ca)axbJcx  and  all  covariants  of  Q 
are  combinants  of  the  three  cubics  a\,  bl,  c&  (Gordan >. 

38.  Let  /  and  g  be  two  binary  forms  of  order  m.  Suppose 
that  <f>  is  any  invariant  of  degree  i  of  a  quantic  of  order  m. 
Then  the  invariant  <f>  constructed  for  the  form  vuf+  v/j  will  be  a 
binary  form  Ft  of  order  i  in  the  variables  vu  v.;.  Prove  that  any 
invariant  of  F{  is  a  combinant  of/,  g.  (Cf.  Salmon,  Lessons  Intro- 
ductory to  Modern  Higher  Algebra,  Fourth  edition,  p.  211.) 

39.  Prove  that  the  Cartesian  equation  of  the  rational  plane 
cubic  curve 

x,  =  ajl  +  aaitt2  +  .-  +  al3B  (i=  1,  2,  3), 
is 

K'o     1  K^O    2*M  ^*0    3*^  I 

$(#!,  x2,  x3)  =    \a0a2x\     |a0o3.x"]  +  [aiO^x'l     la^icl    =0. 
|a0a3x|  la^scl  |«2«3#| 

40.  Show  that  a  binary  quintic  has  two  and  only  two  linearly 
independent  sei  inn  variants  of  degree  five  and  weight  five. 

The  result,  obtained  by  the  annihilator  theory,  is 
\(afa5  —  5  aftalai  +  10  al<i\a3  —  10  a0a%a2  +  4  a|) 

+  /u.(a0a2  —  «i)(tt02a3  —  3  acai02  +  2  af). 

41.  Demonstrate  that  the  number  of  linearly  independent 
semin variants  of  weight  w  and  degree  i  of  a  binary  form  of  order 
m  is  equal  to 

(iv ;  i,  rn)  —  (w  —  1 ;  i,  m), 


APPENDIX  239 

where  (w ;  i,  m)  denotes  the  number  of  different  partitions  of  the 
number  w  into  i  or  fewer  numbers,  none  exceeding  m.  (A  proof 
of  this  theorem  is  given  in  Chapter  VII  of  Elliotts'  Algebra  of 
Qualities.) 

42.  If  /=  a™  =  b™  =  —  is  a  ternary  form  of  order  m,  show  that 

(/,  /)«•  -'*  =  (abuy-ha'>>-2kb'r2k. 
Prove  also 

<«  /)0' " /y- '  -  ?2^m  i  C  ~ 2  *X" -'  *)  w 

X  (a6w)2*-r(&cw)*-,'(acw)*a™~<"2fc&2'_,+<~2*c?~r~*- 

43.  Derive  all  of  the  invariant  formations  of  degrees  1,  2,  3,  4 
of  the  ternary  cubic,  as  given  in  Table  VII,  by  the  process  of  pass- 
ing by  transvection  from  those  of  one  degree  to  those  of  the  next 
higher  degree. 

44.  We  have  shown  that  the  seminvariant  leading  coefficient  of 
the  binary  co variant  of  /=  a™, 

<f>  =  (aby(ac)«  —  aj$?*», 
is 

<£0=  (a6)p(ac)«.-«f6f  •••. 

If  we  replace  av  by  ax,  6X  by  bx,  etc.  in  <f>Q  and  leave  a2,  b2,  ••• 
unchanged,  the  factor  (ab)  becomes 

(alxl  +  a2x2)b2  —  (b^  -f-  b2x2)a2  =  (ab)xv 
At  the  same  time  the  actual  coefficient  ar  =  a™~rar2  of  /  becomes 

\m  —  r  Qrf 


cC-ra2  = 


[m     daS 


Hence,  except  for  a  multiplier  which  is  a  power  of  xu  a  binary 
covariant  may  be  derived  from  its  leading  coefficient  <£0  by  re- 
placing in  <£0,  a0,  a1;  —,  am  respectively  by 

f  1.K  1         d2f    _       \m—rdrf    _       1  d'"f  t 

m  dx2    m{m  —  1)  d.rf  |  m     dx2       '  | m  da™ 

Illustrate  this  by  the  covariant  Hessian  of  a  quartic. 


240 


THE   THEORY   OF  INVARIANTS 


45.  Prove  that  any  ternary  concomitant  of  /=  a™  can  be  de- 
duced from  its  leading  coefficient  (save  for  a  power  of  ux)  by  re- 
placing, in  the  coefficient,  a     by 


mm-- 


(Cf.  Forsyth,  Amer.  Journal  of  Math.,  1889.) 

46.  Derive  a  syzygy  between  the  simultaneous  concomitants  of 
two  binary  quadratic  forms  /,  g  (Chap.  VI). 

The  result  is 

-2J\,  =  Dx?  +  D2r-2hfg, 

where  J12  is  the  Jacobian  of   the  two  forms,  h  their  bilinear  in- 
variant, and  Du  D2  the  respective  discriminants  of  /  and  g. 

47.  Compute  the  transvectant 

of  the  ternary  cubic 

^      |3 
/=  asx  =  bl  =  V . ,_  *T  , , .flWafaflEg, 


\z\i\l 


in  terms  of  its  coefficients  apqT  (p  +  g-)-r  =  3). 

The  result  for  ■£(/,  /)0,  -  is  given  in  the  table  below.  Note  that 
this  mixed  concomitant  may  also  be  obtained  by  applying 
Clebsch's  translation  principle  to  the  Hessian  of  a  binary  cubic. 


4>q 

•''i"i"-j 

•ri"i 

B?Wl«8 

a^UjWj 

a^«| 

O120O102 
-  «m 

2  01110201 

—  2  O210O102 

O102O300 

—  a201 

2  0210am 

—  2  0120^201 

2  O201O210 

—  2  OniO300 

03000120 

~  a210 

aWi 

(B1«aW1«8 

a-jXo"^ 

trjir^WjUg 

"WWs 

"Wa 

01200012 
—  2OinO021 

4-    O102O030 

2oni 

—  2fl2ioOoi2 

—  2O102O120 

+  2  O201O021 

O102O210 
—  2O20lOin 
+       O300«012 

2  O210O021 
—  2  O201O030 

2  O201O120 
—  2  O300O021 

O300O030 

—  O21QO120 

APPENDIX 


241 


a-hi% 

0 i"l"2 

a-|w| 

^""l^S 

JoK2M3 

«•!«! 

00300012 
-  «021 

2  ao2iain 

—  2  «120«012 

00120210 
-  «1U 

2  0!l20«021 

—  2  ao3offiii 

2  aiiiano 
—  2  ao2ifi(2io 

O210«030 
—  a120 

x^siq 

Vl'Vi 

aV>3i4 

«1«!»1«1 

»l«3«2'«8 

cr,3-3«^ 

Ol20«003 

—  2  amaoi2 

+       «102«021 

2  02010012 
—  2  a2io«oo3 

«300«003 
—  O20lOl02 

2«m 

—  2  02310021 

—  2  «l20Ol02 

+  2  a2ioOoi2 

2  «210Ol02 
—  2  «300«012 

O120O2OI 

—  2  aiii#2io 

+     O300O02I 

a-2xtiu\ 

x^r^u^ii 

SSg£B3'u2 

a&Wh 

a'Sa!3M2w3 

WI 

a030«003 

—  00210012 

2  «02iai02 
—  2  ai2o«oo3 

0012^201 

—  2  am«io2 

+      O210O003 

2  «120«012 
—  2  «030Oi02 

2«m 

—  2  ao2i«2oi 

—  2  0210^012 

+  2ai20«i02 

O210O021 

—  2ai2oain 

+      Oo30«201 

4>4 

a-J^jMo 

a|«2 

a^«n^ 

^"s 

a-3«i 

O021O003 
2 
—  afe 

2  00120102 
—  2  fflmaoo3 

O0O3O2OI 
2 
—  a102 

2  aniaoi2 
—  2  00210102 

2  01020111 
—  2  O012O201 

02010021 

-«m 

48.   Prove  that  a  modular  binary  form  of  even  order,  the 
modulus  being  p  >  2,  has  no  covariant  of  odd  order. 

(Suggestion.    Compare  Chap.  II,  §  2,  II.    If  \  is  chosen  as  a  primitive 
root,  equation  (48)  becomes  a  congruence  modulo  p  —  1.) 


„„„,  .  \jrictut;  : 

definition,  23  of  binary  concomitant,  123 

systems,  144-161  of  ternary  concomitant.  220 

universal,  32  Group  : 
Cubic   binary  :  of  transformations,  18 

fundamental  system,  68,  100,  141  the  induced  group,  19 

243 


25 

28 


29 

31 

33 
37 

39 
45 
52 


In  (12);  for  %  read  jt 
For  c/cs.,  read  9/3xa' 
In    the    subscript   of    the 
second   element   of    the 
first  row  read  x.^  for  x2 
For      i  n"  (n  —  1)"      read 

i»i»(m-l)»  .  .  . 
For  A0,  Am  read  a0,  am 
For  (2,  2)  read  (2.  3)  . 
For  w  =  w  read  w  =§  w  . 
Kead  (Jr  Xo)u  for  (x15  ./•.,) 
For  a'=/^,„  read  a,;=/0jU 
For  (-  l)r  read  (-  1)*  . 


ERRATA 

LINE 

PAGE 

.   30 

119 

.  18 

122 

137 

137 

,  17 

142 

i 

145 

5 

158 

(>.  17 

.  12 

100 

.  15 

184 

.  16 

187 

22 


LINE 

Delete  Kt,  •••     .     .     .     .  6 

For  /x  =*■  j-J  n  read  n  s  |  „   .  ^  1 

For  C|  read  C2*      .     .     .  24 

For  <p  read  0 28 

For   Aj  read  hf    .     .     .     .  5 

For  2  <t  ="  e  read  2<t2(  .  28 
For  <f>g~'2  read  0.fj"1.  and 

in  line 26  read  A„,  for.!,,  is 

In  lix  and  7i.,  read  m  for  n  13 
Read  -  #5  for  -  (W  .  .  6,  8 
For  (X/tw):=0  read  (X/w^O  26 
The    form    of    degree    1 1 

should  read  (a<ju)a;/j*  .  16 


INDEX 


Absolute  covariants,  2,  42 
Algebraically  complete  systems,  see 

fundamental  systems 
Anharmonic  ratio,  3 
Annihilators  : 

binary,  25 

ternary,  189 
Anti-seminvariants,  176,  179 
Apolarity,  51,  173 
Arithmetical  invariants,  12,  32,  48, 

157 
Aronhold's  polar  operators,  46 
Associated  forms,  158 

Bezout's  resultant,  168 
Bilinear  invariants,  51 
Boolean  concomitants : 

of  a  linear  form,  156 

of  a  quadratic,  157 

Canonical  forms  : 

binary  cubic,  108 

binary  quartic,  111 

binary  sextic,  112 

ternary  cubic,  111 
Class  of  ternary  form,  230 
Classes  in  modular  theory,  204 
Cogrediency,  20 
Combinants,  162 
Complete  systems  : 

absolutely,  129 

relatively,  130  _       

Conic,  system  of,  224  ^, 

Contragrediency,  212 
Contravariants,  228 
Conversion  operators,  70 
Convolution,  93,  220 
Coordinates,  15 
Covariant  curves,  171 
Covariants  : 

definition,  23 

systems,  144-161 

universal,  32 
Cubic,  binary  : 

fundamental  system,  68,  100,  141 


Cubic,  binary  : 

canonical  form,  108 

syzygy,  107,  110,  161 
Cubic,  ternary  : 

fundamental  system,  225 

canonical  form,  111 

semi-discriminants,  193 

Degree,  20 

Determinant,  symbolical,  55,  170 

Differential  equation  : 

satisfied  by  combinants,  163 

(see  also  annihilators) 
Differential  invariant,  9 
Diophantine  equations,  116 
Discriminant,  4,  31 

Eliminant,  30 
End  coefficients,  179 
Euler's  theorem,  44 
Existence  theorem,  40 

Factors  of  forms,  69,  191 
Fermat's  theorem,  14,  21 
Finiteness : 

algebraical  concomitants,  66 
formal-modular      concomitants, 

204 
modular  concomitants,  204 
Formal   modular   concomitants,    12, 

157 
Fundamental  systems,  144,  161,  204, 
223,  225 

Geometry  of  point  ranges,  78 
Gordan's  proof  of  Hilbert's  theorem, 

112 
Gordan's  series,  83 
Gordan's  theorem,  128 
Grade : 

of  binary  concomitant,  123 

of  ternary  concomitant.  220 
Group  : 

of  transformations,  18 

the  induced  group,  19 


243 


244 


THE  THEORY  OF  INVARIANTS 


Harmonically  conjugate,  6 
Hermite's  reciprocity  theorem,  76 
Hesse's  canonical  form,  111 
Hessians,  28 
Hilbert's  theorem,  112 

Identities,  fundamental : 

binary,  66 

ternary,  218 
Index,  34 
Induced  group,  19 
Inflexion  points,  171 
Intermediate  concomitants,  47 
Invariant  area,  1 
Invariants : 

fundamental  systems,  144-161 

modular,  203 

formal  modular,  157,  204 
Involution,  78 
Irreducible  systems,  see  fundamental 

systems 
Isobarism,  35 

Jacobians,  27 
Jordan's  lemma,  119 

Line  equation  : 

of  conic,  223 

of  cubic,  229 

of  form  of  order  m,  230 
Linear  transformations,  15 
Linearly  independent  seminvariants, 
178,  205. 

Mixed  concomitants,  228 
Modular : 

concomitants,  203 

forms,  203 

transformation,  12 

Operators 

{see  annihilators) 
conversion,  70 
Aronhold,  46 

Parametric  representation,  169 

Partitions,  238 

Polars,  42 

Projective  properties,  78 

Quadratic,  65 
Quadric,  225 
Quartic,  89 


Quaternary  form,  33 
Quintic,  147 

Range  of  points,  78 

Rational  curves,  169 

Reciprocity,  Hermite's  law,  76 

Reduction,  64,  83 

Representation,  typical,  159 

Resultants,  29,  166,  168 

Resultants   in   Aronhold's   symbols, 

151 
Robert's  theorem,  179 
Roots,  concomitants  in  terms  of,  69 

Semi-discriminants,  193 
Seminvariants  : 

algebraic,  175 

modular,  205 
Sextic,  canonical  form  of,  112 
Simultaneous  concomitants,  23 
Skew  concomitants,  39 
Standard  method  of  transvection: 

binary,  67 

ternary,  219 
Stroh's  series,  89 
Symbolical  theory  : 

binary,  53 

ternary,  209 
Symmetric  functions  : 

binary,  69 

ternary,  191 
Symmetry,  39 
Syzygies  : 

algebraic,  104 

modular,  208 

Tables  : 

I.  Concomitants  of  binary  cubic, 
68 

II.  Concomitants  of  binary  quar- 
tic, 89 

III.  System   of    quadratic    and 
cubic,  147 

IV.  System  of  quintic,  150 

V.  Semi-discriminants    of     ter- 
nary cubic,  200 

VI.  Modular   system    of  quad- 
ratic, 204 

VII.  System   of  ternary  cubic, 
225 

Ternary  qualities : 

symbolical  theory,  209 
transvection,  219 
fundamental  systems,  223,  225 


INDEX 


245 


Transformations,  non-linear,  9 

(see  linear  transformations) 
Transformed  form  : 

binary,  16 

ternary,  187 
Translation  principle : 

Clebsch's,  228 

Meyer's,  169 
Transvectants,  binary : 

Definition,  51 

Theorems  on,  92 


Transvectants,  ternary  : 

Definition,  209,  219 

Theorems  on,  220 
Types,  48 
Typical  representation,  159 

Uniqueness  of  canonical  reduction, 

109,  112 
Universal  covariants,  32,  212 

Weight,  34 


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_ 


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MAY  2  9 1982 


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